Published for SISSA by
Springer
Received: December 17,
Revised: March 30,
Accepted: April 2,
Published: May 4,
2011
2012
2012
2012
M. Arana-Catania,a S. Heinemeyer,b M.J. Herreroa and S. Pe˜
narandac
a
Departamento de F´ısica Te´
orica and Instituto de F´ısica Te´
orica,
IFT-UAM/CSIC, Universidad Aut´
onoma de Madrid,
Nicol´
as Cabrera 13–15, Cantoblanco, E-28049 Madrid, Spain
b
Instituto de F´ısica de Cantabria (IFCA-UC/CSIC),
Avda. de los Castros s/n, E-39005 Santander, Spain
c
Departamento de F´ısica Te´
orica, Universidad de Zaragoza,
Pedro Cerbuna 12, E-50009 Zaragoza, Spain
E-mail: [email protected], [email protected],
[email protected], [email protected]
Abstract: We present one-loop corrections to the Higgs boson masses in the MSSM
with Non-Minimal Flavor Violation. The flavor violation is generated from the hypothesis
of general flavor mixing in the squark mass matrices, and these are parametrized by a
XY (X, Y = L, R; i, j = t, c, u or b, s, d). We calculate the corrections to
complete set of δij
XY taking into account all relevant restrictions from
the Higgs masses in terms of these δij
B-physics data. This includes constraints from BR(B → Xs γ), BR(Bs → µ+ µ− ) and
∆MBs . After taking into account these constraints we find sizable corrections to the Higgs
boson masses, in the case of the lightest MSSM Higgs boson mass exceeding tens of GeV.
These corrections are found mainly for the low tan β case. In the case of a Higgs boson
XY .
mass measurement these corrections might be used to set further constraints on δij
Keywords: Supersymmetry Phenomenology
ArXiv ePrint: 1109.6232
c SISSA 2012
doi:10.1007/JHEP05(2012)015
JHEP05(2012)015
Higgs boson masses and B-physics constraints in
Non-Minimal Flavor Violating SUSY scenarios
Contents
1 Introduction
1
2 SUSY scenarios with Non-Minimal Flavor Violation
3
4 Radiative corrections to MSSM Higgs masses within NMFV scenarios
4.1 The Higgs boson sector at tree-level
4.2 The Higgs boson sector at one-loop
4.3 Analytical results of Higgs mass corrections in NMFV-SUSY
21
21
25
27
5 Numerical analysis
XY =
5.1 ∆mφ versus one δij
6 0
XY
6 0
5.2 ∆mh versus two δij =
28
28
29
6 Conclusions
36
A Feynman rules
40
B Tadpoles and self-energies
44
1
Introduction
The Higgs sector of the Minimal Supersymmetric Standard Model (MSSM) [1–3] with two
scalar doublets accommodates five physical Higgs bosons. In lowest order these are the
light and heavy CP-even h and H, the CP-odd A, and the charged Higgs bosons H ± . The
Higgs sector of the MSSM can be expressed at lowest order in terms of the gauge couplings,
MA and tan β ≡ v2 /v1 , the ratio of the two vacuum expectation values. All other masses
and mixing angles can therefore be predicted. Higher-order contributions can give large
corrections to the tree-level relations (see e.g. ref. [4, 5] for reviews).
The MSSM predicts scalar partners for each fermionic degree of freedom in the Standard Model (SM), and fermionic partners for Higgs and gauge bosons. So far, the direct
search for SUSY particles has not been successful. One can only set lower bounds of several hundreds of GeV, depending on the particle and the model specifications [6, 7]. To lift
–1–
JHEP05(2012)015
3 Constraints on Non-Minimal Flavor Violating SUSY scenarios from B8
physics
3.1 BR(B → Xs γ)
9
+
−
3.2 BR(Bs → µ µ )
11
3.3 ∆MBs
12
3.4 Numerical results on B observables
14
1
See [11] and references therein.
We concentrate here on the case with real parameters. For complex parameters see refs. [22, 23] and
references therein.
2
–2–
JHEP05(2012)015
the masses of the SUSY partners from the corresponding SM values, soft SUSY-breaking
terms are introduced [1–3]. The most general flavor structure of the soft SUSY-breaking
sector with flavor non-diagonal terms would induce large flavor-changing neutral-currents,
contradicting the experimental results [8]. Attempts to avoid this kind of problem include
flavor-blind SUSY-breaking scenarios, like minimal Supergravity or gauge-mediated SUSYbreaking. In these scenarios, the sfermion-mass matrices are flavor diagonal in the same
basis as the quark matrices at the SUSY-breaking scale. However, a certain amount of flavor
mixing is generated due to the renormalization-group evolution from the SUSY-breaking
scale down to the electroweak scale. In a more agnostic approach all flavor-violating terms
are introduced as free parameters, and each model point, i.e. each combination of flavorviolating parameters, is tested against experimental data.
Similarly strong bounds exist for the MSSM Higgs sector from the non-observation of
Higgs bosons at LEP [9, 10], the Tevatron1 and most recently from LHC searches [12]. The
LHC has good prospects to discover at least one neutral Higgs boson over the full MSSM
parameter space. A precision on the mass of a SM-like Higgs boson of ∼ 200 MeV is expected [13–15]. At the ILC a determination of the Higgs boson properties (within the kinematic reach) will be possible, and an accuracy on the mass could reach the 50 MeV level [16–
19]. The interplay of the LHC and the ILC in the neutral MSSM Higgs sector is discussed
in ref. [20, 21].
For the MSSM2 the status of higher-order corrections to the masses and mixing angles
in the neutral Higgs sector is quite advanced. The full one-loop and potentially all leading
two-loop corrections are known, see ref. [24] for a review. Most recently leading three-loop
corrections became available [25–27].
However, the impact of non-minimal flavor violation (NMFV) on the MSSM Higgsboson masses and mixing angles, entering already at the one-loop level, has not been
explored very deeply so far. A one-loop calculation taking into account the LL-mixing
between the third and second generation of scalar up-type quarks has been performed in
ref. [28]. A full one-loop calculation of the Higgs-boson self-energies including all NMFV
mixing terms had been implemented into the Fortran code FeynHiggs [23, 24, 29–31],
however no cross checks or numerical evaluations analyzing the effects of the new mixing
terms were performed. Possible effects from NMFV on Higgs boson decays were investigated in [32–36]. Within a similar context of NMFV there have been also studied some
effects of scharm-stop flavor mixing in top-quark FCNC processes [37] and charged Higgs
processes [38] as well as the implications for LHC [38, 39]. Some previous studies on the
induced radiative corrections on the Higgs mass from scharm-stop flavor mixing have also
been performed in [37, 39], but any effects from mixing involving the first generation of
scalar quarks have been neglected. The numerical estimates in [37, 39] also neglect all the
flavor mixings in the scalar down-type sector, except for those of LL-type that are induced
from the scalar up-type sector by SU(2) invariance. In [39] they also consider one example
with a particular numerical value of non-vanishing s˜L − ˜bR mixing.
2
SUSY scenarios with Non-Minimal Flavor Violation
We work in SUSY scenarios with the same particle content as the MSSM, but with general flavor mixing hypothesis in the squark sector. Within these SUSY-NMFV scenarios,
besides the usual flavor violation originated by the CKM matrix of the quark sector, the
general flavor mixing in the squark mass matrices additionally generates flavor violation
from the squark sector. These squark flavor mixings are usually described in terms of a set
XY (X, Y = L, R; i, j = t, c, u or b, s, d), which for simplicity
of dimensionless parameters δij
in the computations are frequently considered within the Mass Insertion Approximation
(MIA) [48]. We will not use here this approximation, but instead we will solve exactly the
diagonalization of the squark mass matrices.
3
Subleading NLO MSSM corrections were evaluated in [40, 41] . However, their effect on our evaluations
would be minor.
4
See [42, 43] and references therein.
–3–
JHEP05(2012)015
We study in this paper the consequences from NMFV for the MSSM Higgs-boson spectrum, where our results are obtained in full generality, i.e. all generations in the scalar upand down-type quark sector are included in our analytical results. In the numerical analysis we focus particularly on the flavor mixing between the second and third generations
which is the relevant one in B physics. Our estimates include all type of flavor mixings,
LL, LR, RL, and RR. We devote special attention to the LR/RL sector. These kind of
mixing effects are expected to be sizable, since they enter the off-diagonal A parameters,
which appear directly in the coupling of the Higgs bosons to scalar quarks.
Concerning the constraints from flavor observables we take into account the most
up-to-date evaluations in the NMFV MSSM for BR(B → Xs γ),3 BR(Bs → µ+ µ− ) and
∆MBs , based on the BPHYSICS subroutine included in the SuFla code.4 For the evaluation of ∆MBs we have incorporated into this subroutine additional contributions from the
one-loop gluino boxes [44] which are known to be very relevant in the context of NMFV
scenarios [45–47].
In the first step of the analysis we scan over the NMFV parameters, and contrast
them with the experimental bounds on BR(B → Xs γ), BR(Bs → µ+ µ− ) and ∆MBs .
In the second step we analyze the one-loop contributions of NMFV to the MSSM Higgs
boson masses, focusing on the parameter space still allowed by the experimental flavor
constraints. In this way the full possible impact of NMFV in the MSSM on the Higgs
sector can be explored. The paper is organized as follows: in section 2 we introduce our
notation for the NMFV MSSM and define certain benchmark scenarios that are used for
the subsequent analysis. Our implementation and new results on B-physics observables is
given in section 3, where we also analyze in detail which combination of NMFV parameters
are still allowed by current experimental constraints. The calculation of the corrections to
Higgs boson masses in the NMFV MSSM is presented in section 4. The numerical analysis
of the impact of the one-loop Higgs mass corrections is given in section 5. Our conclusions
can be found in section 6.
In this section we summarize the main features of the squark flavor mixing within the
SUSY-NMFV scenarios and set the notation.
The relevant MSSM superpotential terms are:
h
i
ˆ 2α Q
ˆ β Y uU
ˆ −H
ˆ 1α Q
ˆ β Y dD
ˆ + µH
ˆ 1α H
ˆβ ,
W = ǫαβ H
(2.1)
2



uphys
uint
L,R
L,R
 phys 

u 
 cL,R  = VL,R  cint
L,R  ,
tint
tphys
L,R
L,R




dphys
dint
L,R
L,R
 phys 

d 
 sL,R  = VL,R  sint
L,R  ,
bint
bphys
L,R
L,R

(2.2)
such that the fermion mass matrices in the physical basis are diagonal:
mu mc mt
VLu Y u∗ VRu† = diag(yu , yc , yt ) = diag
,
,
,
(2.3)
v2 v2 v2
md ms mb
d d∗ d†
,
,
VL Y VR = diag(yd , ys , yb ) = diag
,
(2.4)
v1 v1 v1
where v1 = H10 and v2 = H20 are the vacuum expectation values of the neutral Higgs
fields. The CKM matrix, which is responsible for the flavor violation in the quark sector,
is given as usual as,
(2.5)
VCKM = VLu VLd† .
A simultaneous (parallel) rotation of the squarks with the same above unitary matrices as
their corresponding quark partners leads to the so-called Super-CKM basis. In the NMFV
scenarios, contrary to the MFV ones, the Super-CKM basis does not coincide with the
physical squark basis, i.e, their corresponding squark mass matrices are not yet diagonal.
int to the Super-CKM
More concretely, rotating the squarks from the interaction basis, q˜L,R
basis, q˜L,R , by








˜int
˜L,R
u
˜int
d
u
˜L,R
d
L,R
L,R






u 
d 
(2.6)
 c˜L,R  = VL,R  c˜int
 s˜L,R  = VL,R  s˜int
L,R  ,
L,R  ,
˜bL,R
˜bint
t˜L,R
t˜int
L,R
L,R
one gets the soft-SUSY-breaking Lagrangian transformed from:
˜ iint∗ m2˜ D
˜ int − Q
˜ int† m2˜ Q
˜ int
Lsoft = − U˜iint∗ m2U˜ ij U˜jint − D
i
Dij j
Qij j
i
h
˜ int
¯u ˜int∗ H2 − Q
˜ int
¯d ˜ int∗
− Q
i Aij Uj
i Aij Dj H1 + h.c.
–4–
(2.7)
JHEP05(2012)015
ˆ containing the quark (uL dL )T and squark (˜
where the involved superfields are: Q,
uL d˜L )T
ˆ , containing the quark (uR )c and squark u
ˆ containing
SU(2) doublets; U
˜∗R SU(2) singlets; D,
ˆ 1,2 containing the Higgs bosons SU(2)
the quark (dR )c and squark d˜∗R SU(2) singlets; and H
−
+
doublets, H1 = (H10 H1 )T and H2 = (H2 H20 )T , and their SUSY partners. f c denotes
here the particle-antiparticle conjugate of fermion f , and f˜∗ denotes the complex conjugate
of sfermion f˜. We follow the notation of [49], but with the the convention ǫ12 = −1. The
ˆ i ,U
ˆi , D
ˆ i , quarks qi , squarks q˜i , (i = 1, 2, 3), and
generation indices in the superfields, Q
Yukawa coupling 3×3 matrices, Yiju , Yijd , (i, j = 1, 2, 3), have been omitted above for brevity.
Usually one starts rotating the quark fields from the SU(2) (interaction) eigenstate
int , to the mass (physical), q phys eigenstate basis by unitary transformations, V u,d :
basis, qL,R
L,R
L,R
to:
∗
∗
∗
∗
˜ Ri
˜ Rj − U˜Li
˜ Li
˜ Lj
Lsoft = −U˜Ri
m2U˜R ij U˜Rj − D
m2D˜ R ij D
m2U˜L ij U˜Lj − D
m2D˜ L ij D
(2.8)
h
i
∗
˜ Li (VCKM )ki Au U˜∗ H+ − U˜Li (V ∗ )ik Ad D
˜ ∗ H− + D
˜ Li Ad D
˜ ∗ H0 +h.c. ,
− U˜Li Auij U˜Rj
H20 − D
kj Rj 2
CKM
kj Rj 1
ij Rj 1
where we have used calligraphic capital letters for the squark fields with generation indexes,
and (q = u, d)
Aq = VLq A¯q VRq† , m2U˜ = VRu m2U˜ VRu† , m2D˜ = VRd m2D˜ VRd† , m2U˜ = VLu m2Q˜ VLu† , m2D˜ = VLd m2Q˜ VLd† .
R
R
L
L
(2.11)
The usual procedure to introduce general flavor mixing in the squark sector is to include the
non-diagonality in flavor space at this stage, namely, in the Super-CKM basis. Thus, one
usually writes the 6 × 6 non-diagonal mass matrices, M2u˜ and M2d˜, referred to the SuperCKM basis, being ordered respectively as (˜
uL , c˜L , t˜L , u
˜R , c˜R , t˜R ) and (d˜L , s˜L , ˜bL , d˜R , s˜R , ˜bR ),
and write them in terms of left- and right-handed blocks Mq˜2AB (q = u, d, A, B = L, R),
which are non-diagonal 3 × 3 matrices,
M2q˜
=
Mq˜2LL Mq˜2LR
†
Mq˜2RR
Mq˜2LR
!
,
˜
q˜ = u
˜, d.
(2.12)
where:
Mu˜2 LL ij =m2U˜
Mu˜2 RR ij
Mu˜2 LR ij
Md2˜LL ij
Md2˜RR ij
Md2˜LR ij
+ m2ui + (T3u − Qu sin2 θW )MZ2 cos 2β δij ,
=m2U˜ ij + m2ui + Qu sin2 θW MZ2 cos 2β δij ,
R
= H20 Auij − mui µ cot β δij ,
=m2D˜ ij + m2di + (T3d − Qd sin2 θW )MZ2 cos 2β δij ,
L
2
=mD˜ ij + m2di + Qd sin2 θW MZ2 cos 2β δij ,
R
= H10 Adij − mdi µ tan β δij ,
L
ij
(2.13)
with, i, j = 1, 2, 3, Qu = 2/3, Qd = −1/3, T3u = 1/2 and T3d = −1/2. θW is the weak angle,
MZ is the Z gauge boson mass, and (mu1 , mu2 , mu3 ) = (mu , mc , mt ), (md1 , md2 , md3 ) =
(md , ms , mb ). It should be noted that the non-diagonality in flavor comes from the values
of m2U˜ ij , m2U˜ ij , m2D˜ ij , m2D˜ ij , Auij and Adij for i 6= j.
L
R
L
R
The next step is to rotate the squark states from the Super-CKM basis, q˜L,R ,
to the physical basis, q˜phys . If we set the order in the Super-CKM basis as above,
(˜
uL , c˜L , t˜L , u
˜R , c˜R , t˜R ) and (d˜L , s˜L , ˜bL , d˜R , s˜R , ˜bR ), and in the physical basis as u
˜1,...6 and
–5–
JHEP05(2012)015
int
˜int ˜int , ˜bint ; Q
˜int T cint s˜int )T , (t˜int˜bint )T ; (2.9)
˜ int
˜int ˜ int
U˜1,2,3
=u
˜int
˜int
uint
R ,c
R , tR ; D1,2,3 = dR , s
R
R
1,2,3 = (˜
L dL ) , (˜
L L
L L
˜ L1,2,3 = d˜L , s˜L , ˜bL ; U˜R1,2,3 = u
˜ R1,2,3 = d˜R , s˜R , ˜bR ;
U˜L1,2,3 = u
˜L , c˜L , t˜L ; D
˜R , c˜R , t˜R ; D
(2.10)
˜
d˜1,...6 , respectively, these last rotations are given by two 6 × 6 matrices, Ru˜ and Rd ,



u
˜L
u
˜1
 c˜L 

u


 ˜2 
 t˜ 

u

 ˜3 
L 
,
  = Ru˜ 
u
u
˜ 
˜ 
 R
 4
 c˜R 
u
˜5 
t˜R
u
˜6

˜ 
˜ 
dL
d1
 s˜L 
 d˜2 


 
 ˜b 
 d˜ 
˜

 3
L 
 ˜  = Rd  ˜  ,
 dR 
 d4 


 
 s˜R 
 d˜5 
˜bR
d˜6
(2.14)
yielding the diagonal mass-squared matrices as follows,
(2.15)
.
(2.16)
diag{m2d˜ , m2d˜ , m2d˜ , m2d˜ , m2d˜ , m2d˜ }
1
2
3
4
5
6
=R
d˜
˜
M2d˜ Rd†
The corresponding Feynman rules in the physical basis for the vertices needed for
our applications, i.e. the interaction of one and two Higgs or gauge bosons with two
squarks, can be found in the appendix A. This new set of generalized vertices had been
implemented into the program packages FeynArts/FormCalc [50, 51, 53]5 extending the
previous MSSM model file [54]. The extended FeynArts version was used for the evaluation
of the Feynman diagrams along this paper to obtain the general analytical results.
In the numerical part of the present study we will restrict ourselves to the case where
there are flavor mixings exclusively between the second and third squark generation.
These mixings are known to produce the largest flavor violation effects in B meson physics
since their size are usually governed by the third generation quark masses. On the other
hand, and in order to reduce further the number of independent parameters, we will focus
in the following analysis on constrained SUSY scenarios, where the soft mass parameters
fulfill universality hypothesis at the gauge unification (GUT) scale. Concretely, we will
work with the so-called Constrained MSSM (CMSSM) and Non Universal Higgs Mass
(NUHM) scenarios, which are defined by(see [56] and references therein),
CMSSM :
m0 , m1/2 , A0 , sign(µ), tan β
NUHM :
m0 , m1/2 , A0 , sign(µ), tan β, mH1 , mH2 ,
(2.17)
where, A0 is the universal trilinear coupling, m0 , m1/2 , mH1 , mH2 , are the universal scalar
mass, gaugino mass, and H1 and H2 Higgs masses, respectively, at the GUT scale, sign(µ)
is the sign of the µ parameter and again tan β = v2 /v1 . The soft Higgs masses in the
NUHM are usually parametrized as m2H1,2 = (1 + δ1,2 )m20 , such that by taking δ1,2 = 0
one moves from the NUHM to the CMSSM.
It should be noted that the condition of universal squark soft masses,
2
mU˜ ij = m2U˜ ij = m2D˜ ij = m2D˜ ij = m20 δij , is fulfilled just at the GUT scale. When runL
R
L
R
ning these soft mass matrices from the GUT scale down to the relevant low energy, they will
generically turn non-diagonal in flavor. However, in MFV scenarios the non-diagonal terms
are exclusively generated in this running by off-diagonal terms in the VCKM , and therefore
5
The program and the user’s guide are available via [52].
–6–
JHEP05(2012)015
diag{mu2˜1 , m2u˜2 , mu2˜3 , m2u˜4 , m2u˜5 , mu2˜6 } = Ru˜ M2u˜ Ru˜† ,
they can be safely neglected at low energy. Contrary, in NMFV scenarios, the universal
hypothesis in these squark mass matrices is by definition not fulfilled at low energies.
Our final settings for the numerical evaluation of the squark flavor mixings in NMFV
scenarios are fixed (after RGE running) at low energy as follows,


m2U˜ = 
L
m2U˜
L11
0
0
0
m2U˜
L22
LL m
δ23
˜L22 mU
˜L33
U


LL m
δ23
˜L22 mU
˜L33 
U
m2U˜
L33

0
0
0


LR m
0
δct
v 2 Au =  0
˜L22 mU
˜R33 
U
RL m
0 δct
mt At
˜R22 mU
˜L33
U
 2

mU˜
0
0
R11


RR m
m2U˜
δct
m2U˜ =  0
˜R22 mU
˜R33 
U
R22
R
RR m
0 δct
m2U˜
˜R22 mU
˜R33
U
(2.18)
(2.19)
(2.20)
R33
m2D˜
L
=
†
VCKM
m2U˜ VCKM
L


0
0
0


LR
0
δsb mD˜ L22 mD˜ R33 
v1 A =  0
RL m
0 δsb
mb Ab
˜ R22 mD
˜ L33
D
 2

mD˜
0
0
R11


RR m
m2D˜
δsb
m2D˜ =  0
˜ R22 mD
˜ R33 
D
R22
R
RR m
0
δsb
m2D˜
˜ R22 mD
˜ R33
D
d
(2.21)
(2.22)
(2.23)
R33
It is worth mentioning that the relation between the two soft squark mass matrices in the
’Left’ sector (2.21) is due to SU(2) gauge invariance. Eq. (2.21) can be derived from the
two last relations of eq. (2.11). This dependence between the non-diagonal terms of these
LL instead of two independent
squark mass matrices is the reason why is introduced δ23
LL
LL
deltas δct and δsb . To get the needed running of the soft parameters from the GUT scale
down to low energy, that we set here 1 TeV, we solve numerically the one-loop RGEs with
the code SPHENO [57]. The diagonalization of all the mass matrices is performed with
the program FeynHiggs [23, 24, 29–31].
In CMSSM and other SUSY-GUT scenarios the flavor changing deltas go (in the
2
2
LL ≃ − 1 (3m0 +A0 ) (Y q† Y q ) log( MGUT ) (m
˜ 2 is
leading logarithmic approximation) as δ23
23
2
2
MEW
8π
m
˜
usually taken as the geometric mean of the involved flavor diagonal squared squark mass
matrix entries, see eq. (2.24)), whereas the LR, RL and RR deltas are suppressed instead
(m A )
(m2 )
q
q 0
and ∼ m
, respectively. Furthermore, in these scenarios
by small mass ratios, ∼ m
˜2
˜2
the mixing involving the first generation squarks is naturally suppressed by the smallness
of the corresponding Yukawa couplings. In order to keep the number of free parameters
manageable, this motivated our above choice of neglecting in the numerical analysis the
mixing of the first generation squarks. However, we will not assume any explicit hierarchy
in the various mixing terms between the second and third generation.
–7–
JHEP05(2012)015

0
It should be noted that in the ’Left-Right’ sector we have normalized the trilinear
couplings at low energies as Aqij = yqi Aqij (with Au33 = At and Ad33 = Ab ) and we have
neglected the Ai couplings of the first and second generations. Finally, it should be noted
XY defining the non-diagonal entries in flavor space
that the dimensionless parameters δij
(i 6= j) are normalized respect the geometric mean of the corresponding diagonal squared
soft masses. For instance,
LL
δ23
= m2U˜
L23
/(mU˜L22 mU˜L33 ),
R23
/(mU˜R22 mU˜R33 ),
RL
δct
= (v2 Au )32 /(mU˜R22 mU˜L33 ),
etc.
(2.24)
For definiteness and simplicity, in the present work we will perform our estimates in
specific constrained SUSY scenarios, CMSSM and NUHM, whose input parameters m0 ,
XY
m1/2 , A0 , tan β, sign(µ), δ1,2 , are summarized in table 1,6 and supplemented with δij
as described above. Regarding CMSSM, we have chosen six SPS benchmark points [58],
SPS1a, SPS1b, SPS2, SPS3, SPS4, and SPS5 and one more point with very heavy spectra
(VHeavyS). It should be noted that several of these SPSX points are already in conflict
with recent LHC data [6, 7], but we have chosen them here as reference points to study the
effects of SUSY on the Higgs mass corrections, since they have been studied at length in
the literature. At present, a heavier SUSY spectrum, as for instance our point VHeavyS is
certainly more realistic and compatible with LHC data. In general an analysis of LHC data
including NMFV effects in the squark sector would be very desirable. Regarding NUHM,
we have chosen a point with heavy SUSY spectra and light Higgs sector (HeavySLightH)
and a point (BFP) that has been proven in [59] to give the best fit to the set of low energy
observables. For later reference, needed in our posterior analysis of the Higgs mass corrections, we also include in the table the corresponding MSSM Higgs masses, computed with
XY = 0.
FeynHiggs [23, 24, 29–31] and with all flavor changing deltas switched off, i.e., δij
3
Constraints on Non-Minimal Flavor Violating SUSY scenarios from
B-physics
In this section we analyze the constraints on Non-Minimal Flavor Violating SUSY scenarios
from B-Physics. Since we are mainly interested in the phenomenological consequences of
the flavor mixing between the third and second generations we will focus7 on the following
three B meson observables: 1) Branching ratio of the B radiative decay BR(B → Xs γ), 2)
¯s mass difference
Branching ratio of the Bs muonic decay BR(Bs → µ+ µ− ), and 3) Bs − B
∆MBs . Another B observable of interest in the present context is BR(B → Xs l+ l− ).
However, we have not included this in our study, because the predicted rates in NMFVSUSY scenarios for this observable are closely correlated with those from BR(B → Xs γ)
due to the dipole operators dominance in the photon-penguin diagrams mediating BR(B →
6
We adopt here the definition in terms of the GUT-scale input parameters, while the original definition
in [58] was based on the weak-scale parameters.
7
We have checked that electroweak precision observables, where NMFV effects enter, for instance, via
∆ρ [28], do not lead to relevant additional constraints on the allowed parameter space. Our results on this
constraint are in agreement with ref. [37].
–8–
JHEP05(2012)015
LR
δct
= (v2 Au )23 /(mU˜L22 mU˜R33 ),
RR
δct
= m2U˜
m1/2
m0
A0
tan β
δ1
δ2
mh
mH
MA
mH ±
SPS1 a
250
100
-100
10
0
0
112
394
394
402
SPS1 b
400
200
0
30
0
0
116
526
526
532
SPS2
300
1450
0
10
0
0
115 1443 1443
1445
SPS3
400
90
0
10
0
0
115
573
572
578
SPS4
300
400
0
50
0
0
114
404
404
414
SPS5
300
150
-1000
5
0
0
111
694
694
698
VHeavyS
800
800
-800
5
0
0
120 1524 1524
1526
HeavySLightH
600
600
0
5
BFP
530
110
-370
27
−1.86 +1.86 114
−84.7 −84.7 120
223
219
233
507
507
514
Table 1. Values of m1/2 , m0 , A0 , tan β, δ1 , δ2 and Higgs boson masses mh , mH , MA and mH ±
for the points considered in the analysis. All parameters with mass dimension are in GeV, and
sign(µ) > 0 for all points.
XY parameters from
Xs l+ l− ) decays. It implies that the restrictions on the flavor mixing δij
+
−
BR(B → Xs l l )are also expected to be correlated with those from the radiative decays.
The summary of the relevant features for our analysis of these three B meson
observables is given in the following.
3.1
BR(B → Xs γ)
The relevant effective Hamiltonian for this decay is given in terms of the Wilson coefficients
Ci and operators Oi by:
8
4GF ts∗ tb X
Heff = − √ VCKM
(Ci Oi + Ci′ Oi′ ).
VCKM
2
i=1
(3.1)
Where the primed operators can be obtained from the unprimed ones by replacing L ↔ R.
The complete list of operators can be found, for instance, in [55]. In the context of
SUSY scenarios with the MSSM particle content and MFV, only two of these operators
get relevant contributions, the so-called photonic dipole operator O7 and gluonic dipole
operator O8 given, respectively, by:
e
mb (¯
sL σ µν bR ) Fµν ,
16π 2
g3
mb (¯
sL σ µν T a bR ) Gaµν .
O8 =
16π 2
O7 =
(3.2)
(3.3)
′
We have omitted the color indices here for brevity. Within NMFV also O7,8
have to be
taken into account:
e
mb (¯
sR σ µν bL ) Fµν ,
16π 2
g3
O8′ =
mb (¯
sR σ µν T a bL ) Gaµν .
16π 2
O7′ =
–9–
(3.4)
(3.5)
JHEP05(2012)015
points
– 10 –
JHEP05(2012)015
The Wilson coefficients at the SUSY scale are obtained as usual by the matching
procedure of the proper matrix element being computed from the previous effective
Hamiltonian to the corresponding matrix elements being computed from the SUSY model
operating at that SUSY scale, the NMFV-MSSM in our case. These Wilson coefficients
encode, therefore, the contributions to BR(B → Xs γ) from the loops of the SUSY and
Higgs sectors of the MSSM. The effects from squark flavor mixings that are parametrized
XY , are included in this observable via the squark physical masses and rotation
by the δij
matrices, given in the previous section, that appear in the computation of the matrix
element at the one loop level and, therefore, are also encoded in the Wilson coefficients.
The explicit expressions for these coefficients in the MSSM, in terms of the physical basis,
can be found, for instance, in refs. [60–62]. We have included in our analysis the most
relevant loop contributions to the Wilson coefficients, concretely: 1) loops with Higgs
bosons, 2) loops with charginos and 3) loops with gluinos. It should be noted that, at
one loop order, the gluino loops do not contribute in MFV scenarios, but they are very
relevant (dominant in many cases) in the present NMFV scenarios.
Once the Wilson coefficients are known at the SUSY scale, they are evolved with
the proper Renormalization Group Equations (RGEs) down to the proper low-energy
scale. As a consequence of this running the previous operators mix and the corresponding
Wilson coefficients, C7,8 get involved in the computation of the B → Xs γ decay rate. The
RGE-running is done in two steps: The first one is from the SUSY scale down to the
electroweak scale, and the second one is from this electroweak scale down to the B-physics
scale. For the first step, we use the LO-RGEs for the relevant Wilson coefficients as
in [62] and fix six active quark flavors in this running. For the second running we use the
NLO-RGEs as in [63] and fix, correspondingly, five active quark flavors. For the charged
Higgs sector we use the NLO formulas for the Wilson coefficients as in [64].
The resummation of scalar induced large tan β effects is performed, as usual, by the
effective Lagrangian approach that parametrizes the one-loop generated effective couplings
between the H2 Higgs doublet and the down type quarks in softly broken SUSY models [65].
A necessary condition to take into account all tan β-enhanced terms in flavor changing amplitudes is the diagonalization of the down-type quark mass matrix in the presence of
these effective couplings [66–68]. A summary of this effective Lagrangian formalism for the
resummation of large tan β effects in the three B observables of our interest, within the context of MFV scenarios, can be found in [69]. We follow here the treatment of [70] where the
resummation of large tan β effects via effective Lagrangians is generalized to the case where
the effective d¯iR djL H20 coupling contains also non-minimal sources of flavor mixing. It should
be noted that the most relevant scalar induced large tan β effects for the present work are
those generated by one-loop diagrams with gluino-sbottom and chargino-stop inside the
loops. The large tan β resummation effects and the relevance of other chirally enhanced
corrections to FCNC processes within the NMFV context have recently been studied exhaustively also in [71, 72] (previous studies can be found, for instance, in refs. [73–75]).
The total branching ratio for this decay is finally estimated by adding the new
contributions from the SUSY and Higgs sectors to the SM rate. More specifically, we use
eq. (42) of [63] for the estimate of BR(B → Xs γ) in terms of the ratios of the Wilson
′
(including all the mentioned new contributions) divided by the
coefficients C7,8 and C7,8
SM
corresponding C7,8 in the SM.
For the numerical estimates of BR(B → Xs γ) we use the FORTRAN subroutine
′ which were not included in
BPHYSICS (modified as to include the contributions from C7,8
its original version) included in the SuFla code, that incorporates all the above mentioned
ingredients [42, 43].
Finally, for completeness, we include below the experimental measurement of this
observable [8, 76],8 and its prediction within the SM [77]:
BR(B → Xs γ)SM = (3.15 ± 0.23) × 10
3.2
−4
(3.6)
(3.7)
BR(Bs → µ+ µ− )
The relevant effective Hamiltonian for this process is [78, 79]:
GF α ts∗ tb X
Heff = − √ VCKM
VCKM
(Ci Oi + Ci′ Oi′ ),
2π
i
(3.8)
where the operators Oi are given by:
O10 = (¯
sγ ν PL b) (¯
µγν γ5 µ) ,
′
O10
= (¯
sγ ν PR b) (¯
µγν γ5 µ) ,
OS = mb (¯
sPR b) (¯
µµ) ,
OS′ = ms (¯
sPL b) (¯
µµ) ,
OP = mb (¯
sPR b) (¯
µγ5 µ) ,
OP′ = ms (¯
sPL b) (¯
µγ5 µ) .
(3.9)
We have again omitted the color indices here for brevity.
In this case, the RG running is straightforward since the anomalous dimensions of
the above involved operators are zero, and the prediction for the decay rate is simply
expressed by:
BR(Bs → µ+ µ− ) =
G2F α2 m2Bs fB2 s τBs ts∗ tb 2 q
ˆ 2µ
|VCKM VCKM | 1 − 4m
64π 3
ˆ 2µ F10 |2 ,
× 1 − 4m
ˆ 2µ |FS |2 + |FP + 2m
where m
ˆ µ = mµ /mBs and the Fi are given by
#
"
′ m
CS,P mb − CS,P
s
,
FS,P = mBs
mb + ms
(3.10)
′
F10 = C10 − C10
.
Within the SM the most relevant operator is O10 as the Higgs mediated contributions
to OS,P can be safely neglected. It should be noted that the contribution from O10 to
the decay rate is helicity suppressed by a factor of m
ˆ 2µ since the Bs meson has spin zero.
In contrast, in SUSY scenarios the scalar and pseudo-scalar operators, OS,P , can be very
important, particularly at large tan β & 30 where the contributions to CS and CP from
neutral Higgs penguin diagrams can become large and dominate the branching ratio,
8
We have added the various contributions to the experimental error in quadrature.
– 11 –
JHEP05(2012)015
BR(B → Xs γ)exp = (3.55 ± 0.26) × 10−4
BR(Bs → µ+ µ− )exp < 1.1 × 10−8 (95% CL)
BR(Bs → µ+ µ− )SM = (3.6 ± 0.4) × 10−9
3.3
(3.11)
(3.12)
∆MBs
¯s mixing and, hence, for the Bs /B
¯s mass
The relevant effective Hamiltonian for Bs − B
difference ∆MBs is:
2 X
G2
2
tb∗
ts
VCKM
VCKM
Ci O i .
(3.13)
Heff = F2 MW
16π
i
In the SM the most relevant operator is:
OV LL = (¯bα γµ PL sα )(¯bβ γ µ PL sβ ).
(3.14)
Where we have now written explicitly the color indices. In scenarios beyond the SM, as
the present NMFV MSSM, other operators are also relevant:
O1LR = (¯bα γµ PL sα )(¯bβ γ µ PR sβ ),
OSLL = (¯bα PL sα )(¯bβ PL sβ ),
1
O2LR = (¯bα PL sα )(¯bβ PR sβ ),
OSLL = (¯bα σµν PL sα )(¯bβ σ µν PL sβ ),
2
(3.15)
(3.16)
and the corresponding operators OV RR and OiSRR that can be obtained by replacing
PL ↔ PR in (3.14) and (3.16). The mass difference ∆MBs is then evaluated by taking the
matrix element
¯s |Heff |Bs i|,
∆MBs = 2|hB
– 12 –
(3.17)
JHEP05(2012)015
because in this case the branching ratio grows with tan β as tan6 β. The studies in the
literature of these MSSM Higgs-penguin contributions to BR(Bs → µ+ µ− ) have focused
on both MFV [68, 80, 81] and NMFV scenarios [45, 70, 75, 78]. In both cases the rates for
BR(Bs → µ+ µ− ) at large tan β can be enhanced by a few orders of magnitude compared
with the prediction in the SM, therefore providing an optimal window for SUSY signals.
In the present context of SUSY-NMFV, with no preference for large tan β values, there
are in general three types of one-loop diagrams that contribute to the previous Ci Wilson
coefficients for this Bs → µ+ µ− decay: 1) Box diagrams, 2) Z-penguin diagrams and 3)
neutral Higgs boson H-penguin diagrams, where H denotes the three neutral MSSM Higgs
bosons. In our numerical estimates we have included what are known to be the dominant
contributions to these three types of diagrams [78]: chargino contributions to box and
Z-penguin diagrams and chargino and gluino contributions to H-penguin diagrams.
Regarding the resummation of large tan β effects we have followed the same effective
Lagrangian formalism as explained in the previous case of B → Xs γ. In the case of
contributions from H-penguin diagrams to Bs → µ+ µ− this resummation is very relevant
and it has been generalized to NMFV-SUSY scenarios in [70].
For the numerical estimates we use again the BPHYSICS subroutine included in the
SuFla code [42, 43] which incorporates all the ingredients that we have pointed out above.
Finally, for completeness, we include below the present experimental upper bound for
this observable [82], and the prediction within the SM [83]:
¯s |Heff |Bs i is given by
where hB
¯s |Heff |Bs i =
hB
2 X
G2F
2
2
tb∗
ts
M
m
f
V
V
Pi Ci (µW ) .
Bs Bs
CKM CKM
48π 2 W
(3.18)
i
Here mBs is the Bs meson mass, and fBs is the Bs decay constant. The coefficients
Pi contain the effects due to RG running between the electroweak scale µW and mb as
well as the relevant hadronic matrix element. We use the coefficients Pi from the lattice
calculation [84, 85]:
P1LR = − 1.97,
P2LR =2.50,
P1SLL = − 1.02,
P2SLL = − 1.97.
(3.19)
The coefficients P1V RR , etc., may be obtained from those above by simply exchanging
L ↔ R.
In the present context of SUSY-NMFV, again with no preference for large tan β values,
and besides the SM loop contributions, there are in general three types of one-loop diagrams
¯s mixing: 1) Box diagrams,
that contribute to the previous Ci Wilson coefficients for Bs − B
2) Z-penguin diagrams and 3) double Higgs-penguin diagrams. In our numerical estimates
we have included what are known to be the dominant contributions to these three types of
diagrams in scenarios with non-minimal flavor violation (for a review see, for instance, [45]):
gluino contributions to box diagrams, chargino contributions to box and Z-penguin diagrams, and chargino and gluino contributions to double H-penguin diagrams. As in the previous observables, the total prediction for ∆MBs includes, of course, the SM contribution.
Regarding the resummation of large tan β effects we have followed again the effective
Lagrangian formalism, generalized to NMFV-SUSY scenarios [70], as in the previous
cases of B → Xs γ and Bs → µ+ µ− . It should be noted that, in the case of ∆MBs , the
resummation of large tan β effects is very relevant for the double H-penguin contributions,
which grow very fast with tan β.
For the numerical estimates we have modified the BPHYSICS subroutine included in
the SuFla code [42, 43] which incorporates all the ingredients that we have pointed out
above, except the contributions from gluino boxes. These contributions are known to be
¯s mixing in SUSY scenarios with non-minimal flavor violation [45–
very important for Bs − B
47] and therefore they must be included into our analysis of ∆MBs . Concretely, we have
incorporated them into the BPHYSICS subroutine by adding the full one-loop formulas for
the gluino boxes of [44] to the other above quoted contributions that were already included
in the code. In order to illustrate the relevance of these gluino contributions in our analysis
of ∆MBs , we show in figure 1 each separate contribution as a function of tan β in a particular
LL = δ RR = 0.1, that we have chosen for comparison with [45]. The other flaexample with δ23
sb
vor changing deltas are set to zero, and the other relevant MSSM parameters are set to mq˜ =
√
500 GeV, At = −mq˜, mg˜ = 2mq˜, µ = mq˜, and mA = 300 GeV, as in figure 24 of [45].
We clearly see in figure 1 that it is just in the very large tan β region where double Higgspenguins dominate. For moderate and low tan β values, tan β ≤ 20 (which is a relevant
region for our numerical analysis, see below) the gluino boxes fully dominates the SUSY
corrections to ∆MBs and compete with the SM contributions. Our numerical estimate in
– 13 –
JHEP05(2012)015
P1V LL =0.73,
s
SM
Chargino boxes
Double penguins
Gluino boxes
Total
40
20
0
-20
-40
-60
-80
-100
-120
0
10
20
30
40
50
Tanβ
Figure 1. Relevant contributions to ∆MBs in NMFV-SUSY scenarios as a function of tan β.
They include: SM, Double Higgs penguins, gluino boxes and chargino boxes. The total prediction for ∆MBs should be understood here as ∆M√
Bs = |Total|. The parameters are set to
RR
LL
δ23 = δsb = 0.1,mq˜ = 500 GeV, At = −mq˜, mg˜ = 2mq˜, µ = mq˜, and mA = 300 GeV. The
other flavor changing deltas are set to zero.
this plot is in complete agreement with the results in [45] (see, in particular, figure 24 of
this reference) which analyzed and compared in full detail these corrections. Finally, for
completeness, we include below the experimental measurement of this observable [8],9 and
its prediction within the SM (using NLO expression of [86] and error estimate of [87]):
∆MBs exp = (117.0 ± 0.8) × 10−10 MeV ,
+17.2
∆MBs SM = (117.1−16.4
) × 10−10 MeV .
3.4
(3.20)
(3.21)
Numerical results on B observables
In the following of this section we present our numerical results for the three B observables
in the NMFV-SUSY scenarios and a discussion on the allowed values for the flavor
XY .
changing deltas, δij
The predictions for BR(B → Xs γ), BR(Bs → µ+ µ− ) and ∆MBs versus the various
XY , for the six selected SPS points, are displayed respectively in figures 2, 3 and 4. For this
δij
analysis, we have assumed that just one at a time of these deltas is not vanishing. Results for
two non-vanishing deltas will be shown later. The following 7 flavor changing deltas are conLL , δ LR , δ LR , δ RL , δ RL , δ RR and δ RR ; and we have explored delta values within
sidered: δ23
ct
ct
ct
sb
sb
sb
XY < 1. In all plots, the predictions for δ XY = 0 represent our estimate
the interval −1 < δij
ij
of the corresponding observable in the MFV case. This will allow us to compare easily the
9
We have again added the various contributions to the experimental error in quadrature.
– 14 –
JHEP05(2012)015
Contributions to ∆ MB (ps-1)
60
• BR(B → Xs γ):
- Sensitivity to the various deltas:
LR , δ RL , δ LL , δ RR and δ LR , in all the studied
We find strong sensitivity to δsb
ct
23
sb
sb
points, for both high and low tan β values. The order found from the highest
LR and δ RL the largest, 2) δ LL the
to the lowest sensitivity is, generically: 1) δsb
23
sb
LR and δ RR the next to next, and 4) slight sensitivity to δ RR and δ RL .
next, 3) δct
ct
ct
sb
- Comparing the predictions with the experimental data:
LR , δ RL , δ LL , δ RR and δ LR , we
If we focus on the five most relevant deltas, δsb
ct
23
sb
sb
see clearly that tiny deviations from zero (i.e., deviations from MFV) in these
deltas, and specially in the first three, produce sizeable shifts in BR(B → Xs γ),
and in many cases take it out the experimental allowed band. Therefore, it is
obvious from these plots that BR(B → Xs γ) sets stringent bounds on the deltas
LR , δ RL , δ LL ,
(when varying just one delta), which are particularly tight on δsb
23
sb
RR
LR
δsb , and δct , indeed for all the studied SPS points. There are just two exceptions, where the predicted central values of BR(B → Xs γ) are already outside
the experimental band in the MFV case (all deltas set to zero), and assuming
one of these three most relevant deltas to be non-vanishing, the prediction
moves inside the experimental band. This happens, for instance, in the points
SPS4 and SPS1b that have the largest tan β values of 50 and 30 respectively.
Interestingly, it means that some points of the CMSSM, particularly those with
large tan β values, that would have been excluded in the context of MFV, can
be recovered as compatible with data within NMFV-SUSY scenarios.
XY allowed by data:
- Intervals of δij
In order to conclude on the allowed delta intervals we have assumed that
our predictions of BR(B → Xs γ) within SUSY scenarios have a somewhat larger theoretical error ∆theo (BR(B → Xs γ)) than the SM prediction
As a very conservative value we
∆theo
SM (BR(B → Xs γ)) given in (3.7).
theo
−4
XY value is then
use ∆
(BR(B → Xs γ)) = 0.69 × 10 . A given δij
– 15 –
JHEP05(2012)015
results in the two kind of scenarios, NMFV and MFV. It should be noted also, that some
XY < 1,
of the predicted lines in these plots do not expand along the full interval −1 < δij
and they are restricted to a smaller interval; for some regions of the parameter space a too
large delta value can generate very large corrections to any of the masses, and the mass
squared turns negative. These problematic points are consequently not shown in our plots.
We have also included in the right vertical axis of these figures, for comparison, the
respective SM prediction in (3.7), (3.12), and (3.21). The error bars displayed are the
corresponding SM uncertainties as explained below. The shadowed horizontal bands in
the case of BR(B → Xs γ) and ∆MBs are their corresponding experimental measurements
in (3.6), and (3.20), expanded with 3σexp errors. In the case of BR(Bs → µ+ µ− ) the
shadowed area corresponds to the allowed region by the upper bound in (3.11).
The main conclusions extracted from these figures for the three B observables are
summarized as follows:
considered to be allowed by data if the corresponding interval, defined by
BR(B → Xs γ) ± ∆theo (BR(B → Xs γ)), intersects with the experimental band.
It corresponds to adding linearly the experimental uncertainty and the MSSM
theoretical uncertainty. In total a predicted ratio in the interval
2.08 × 10−4 < BR(B → Xs γ) < 5.02 × 10−4 ,
(3.22)
• BR(Bs → µ+ µ− ):
- Sensitivity to the various deltas:
We find significant sensitivity to the NMFV deltas in scenarios with very large
tan β, as it is the case of SPS4 and SPS1b. This sensitivity is clearly due to the
Higgs-mediated contribution that, grows as tan6 β. The largest sensitivity is to
LL . In the case of SPS4, there is also significant sensitivity to δ LR , δ RR and
δ23
sb
sb
LR . In the SPS1b scenario there is also found some sensitivity to δ LR , δ RR ,
δct
ct
sb
RR and δ LR .
δsb
ct
- Comparing the predictions with the experimental data:
XY | ≤ 1 explored values are allowed by
Figure 3 clearly shows that most of the |δij
experimental data on BR(Bs → µ+ µ− ). It is in the points with very large tan β,
i.e SPS4 and SPS1b, where there are some relevant differences between the MFV
and the NMFV predictions. First, all predictions for MFV scenarios except for
SPS4, are inside the experimental allowed area. In the case of SPS1b, the comLL , but also
parison of the NMFV predictions with data constraints specially δ23
LR , δ RR , δ RR and δ LR . In the case of SPS4 some input non-vanishing values of
δsb
ct
ct
sb
LL , δ LR or δ RR can place the prediction inside the experimental allowed area. In
δ23
sb
sb
LL can be found.
the case of the SPS1a and SPS3 scenarios some constraints for δ23
XY allowed by data:
- Intervals of δij
As in the previous observable, we assume here that our predictions for BR(Bs →
µ+ µ− ) have a slightly larger error as the SM prediction in (3.12), where, however, the theory uncertainty is very small in comparison with the experimental
bound. We choose ∆theo (BR(Bs → µ+ µ− )) = 0.12 × 10−8 . Then, a given
XY value is allowed by data if the predicted interval, defined by BR(B →
δij
s
µ+ µ− ) + ∆theo (BR(Bs → µ+ µ− )), intersects the experimental area. The upper
line of the experimental area in this case is set by the 95% CL upper bound
given in (3.11). It implies that for a predicted ratio to be allowed it must fulfill:
BR(Bs → µ+ µ− ) < 1.22 × 10−8 .
– 16 –
(3.23)
JHEP05(2012)015
is regarded as allowed. Our results for these allowed intervals are summarized
in table 2. In this table we see again that the less constrained parameters by
RL , and δ RR . Therefore, except for the excluded SPS4
BR(B → Xs γ) are δct
ct
XY | larger than O(0.1), and compatible
case, these two deltas can be sizeable, |δij
with BR(B → Xs γ) data.
0.0007
LR
δsb
RR
δct
LL
δ23
0.0005
0.0006
RL
δsb
RR
δsb
SM
0.0004
0.0003
0.0002
-0.5
SPS2
LR
δct
0.0005
LR
δsb
RR
δct
LL
δ23
0
δXY
ij
0.5
δsb
0.0005
δ23
SM
RR
LL
0.0004
0.0003
0.0001
-1
1
-0.5
0
0.0007
RL
δct
RL
δsb
RR
δsb
0.0006
SPS3
LR
δct
δct
δsb
δsb
1
δct
δsb
δ23
SM
0.5
1
0.5
1
RL
RL
LR
RR
RR
SM
0.0004
0.0003
0.0002
0.0005
LL
0.0004
0.0003
0.0002
0.0001
-1
-0.5
0
0.5
0.0001
-1
1
-0.5
0
δXY
ij
δXY
ij
SPS4
LR
δct
0.0006
LR
δsb
RR
δct
LL
δ23
0.0005
0.0007
RL
δct
RL
δsb
RR
δsb
0.0006
SPS5
LR
δct
δct
δsb
δsb
δct
δsb
δ23
SM
LR
RR
SM
BR(B->Xs γ)
0.0007
BR(B->Xs γ)
0.5
δXY
ij
BR(B->Xs γ)
BR(B->Xs γ)
δsb
δct
0.0004
0.0003
0.0002
0.0005
LL
RL
RL
RR
0.0004
0.0003
0.0002
0.0001
-1
-0.5
0
0.5
1
δXY
ij
0.0001
-1
-0.5
0
δXY
ij
Figure 2. Sensitivity to the NMFV deltas in BR(B → Xs γ) for the SPSX points of table 1. The
experimental allowed 3σ area is the horizontal colored band. The SM prediction and the theory
uncertainty ∆theo (BR(B → Xs γ)) (red bar) is displayed on the right axis.
XY intervals are collected in table 2. We conclude
The results for the allowed δij
from this table that, except for scenarios with large tan β ≥ 30, like SPS4 and
XY | larger than O(0.1), and
SPS1b, the size of the deltas can be sizeable, |δij
compatible with BR(Bs → µ+ µ− ) data.
• ∆MBs :
- Sensitivity to the various deltas:
– 17 –
JHEP05(2012)015
0.0006
RL
LR
δsb
0.0002
0.0001
-1
0.0007
SPS1b
RL
LR
δct
δct
RR
BR(B->Xs γ)
0.0006
BR(B->Xs γ)
0.0007
SPS1a
RL
LR
δct
δct
2e-08
1.4e-08
1.2e-08
RL
δsb
RR
δsb
6e-09
4e-09
RR
1.4e-08
1.2e-08
0.5
1
-1
-0.5
0
0.5
1
δXY
ij
2e-08
RL
δct
RL
δsb
RR
δsb
1.8e-08
SPS3
LR
δct
δct
1.6e-08
δsb
δsb
δct
δsb
δ23
SM
LR
RR
1.4e-08
BR(Bs->µ+µ-)
LR
δsb
RR
δct
LL
δ23
0
δXY
ij
SM
LL
1.2e-08
1e-08
8e-09
RL
RR
1e-08
8e-09
6e-09
6e-09
4e-09
4e-09
2e-09
RL
2e-09
-1
-0.5
8e-08
BR(Bs->µ+µ-)
7e-08
6e-08
LR
δsb
RR
δct
LL
δ23
0.5
1
-1
2e-08
RL
δct
RL
δsb
RR
δsb
1.8e-08
SPS5
LR
δct
δct
1.6e-08
δsb
δsb
δct
δsb
δ23
SM
1.4e-08
SM
1.2e-08
5e-08
4e-08
3e-08
0
0.5
1
LR
RR
LL
0.5
1
RL
RL
RR
1e-08
8e-09
6e-09
2e-08
4e-09
1e-08
-1
-0.5
δXY
ij
BR(Bs->µ+µ-)
SPS4
LR
δct
0
δXY
ij
2e-09
-0.5
0
δXY
ij
0.5
1
-1
-0.5
0
δXY
ij
Figure 3. Sensitivity to the NMFV deltas in BR(Bs → µ+ µ− ) for the SPSX points of table 1. The
experimental allowed region by the 95% CL bound is the horizontal colored area. The SM prediction
and the theory uncertainty ∆theo (BR(Bs → µ+ µ− )) (red bar) is displayed on the right axis.
Generically, we find strong sensitivity to most of the NMFV deltas in all the
studied points, including those with large and low tan β values. The pattern
XY differs substantially
of the ∆MBs predictions as a function of the various δij
for each SPS point. This is mainly because in this observable there are two
large competing contributions, the double Higgs penguins and the gluino boxes,
with very different behavior with tan β, as we have seen in figure 1. In the
case of SPS4 with extremely large tan β = 50, the high sensitivity to all deltas
– 18 –
JHEP05(2012)015
-0.5
SPS2
LR
δct
1.6e-08
BR(Bs->µ+µ-)
SM
2e-09
-1
9e-08
δ23
8e-09
4e-09
1e-07
δsb
1e-08
6e-09
2e-09
δsb
δct
LL
1.2e-08
RL
δsb
RR
1.4e-08
SM
8e-09
2e-08
LR
1.6e-08
1e-08
1.8e-08
SPS1b
RL
LR
δct
δct
1.8e-08
BR(Bs->µ+µ-)
LR
δsb
RR
δct
LL
δ23
1.6e-08
BR(Bs->µ+µ-)
2e-08
SPS1a
RL
LR
δct
δct
1.8e-08
is evident in this figure. In the case of SPS5 with low tan β = 5, there is
RR , δ LR and δ RL . Generically, for
important sensitivity to all deltas, except δct
ct
ct
LR , δ RL and δ LL ;
all the studied points, we find the highest sensitivity to 1) δsb
23
sb
RR the next, 3) δ LR the next to next; and 4) the lowest sensitivity is to δ RL
2) δsb
ct
ct
RR . Consequently, these two later will be the less constrained ones.
and δct
- Comparing the predictions with the experimental data:
XY allowed by data:
- Intervals of δij
XY value is allowed by ∆M
We consider again, that a given δij
Bs data if the
theo
predicted interval ∆MBs ± ∆
(∆MBs ), intersects the experimental band. It
corresponds to adding linearly the experimental uncertainty and the theoretical
uncertainty. Given the present controversy on the realistic size of the theoretical
error in the estimates of ∆theo (∆MBs ) in the MSSM (see, for instance, [88]), we
choose a very conservative value for the theoretical error in our estimates, considerably larger than the SM value in (3.21), of ∆theo (∆MBs ) = 51×10−10 MeV.
This implies that a predicted mass difference in the interval
63 × 10−10 < ∆MBs (MeV) < 168.6 × 10−10 ,
(3.24)
is regarded as allowed.
The allowed intervals for the deltas that are obtained with this requirement are
collected in table 2. As we have already commented, the restrictions on the
b-sector parameters from ∆MBs are very strong, and in consequence, there are
LR , δ RL , and δ LL . In the case of δ RR there are
narrow intervals allowed for, δsb
23
sb
sb
indeed sequences of very narrow allowed intervals, which in some cases reduce
to just a single allowed value. The parameters that show a largest allowed
XY |, larger than O(0.1), are δ RR , δ RL and δ LR .
interval, with sizeable |δij
ct
ct
ct
XY intervals
Total allowed δij
XY ,
We finally summarize in table 3 the total allowed intervals for all the NMFV deltas, δij
where now we have required compatibility with the present data of the three considered
B observables, BR(B → Xs γ), BR(Bs → µ+ µ− ) and ∆MBs . It is obvious, from the
previous discussion, that the most restrictive observables are BR(B → Xs γ) and ∆MBs ,
leading to a pattern of allowed delta intervals which is clearly the intersect of their two
corresponding intervals. The main conclusion from this table is that, except for SPS4 (the
point SPS4 is practically excluded), the NMFV deltas in the top-sector can be sizeable
– 19 –
JHEP05(2012)015
In this case, the experimental allowed 3σexp band is very narrow, and all the
XY = 0, i.e. for MFV scenarios, lay indeed outside this
central predictions at δij
band. However, if we assume again that our predictions suffer of a similar large
uncertainty as the SM prediction, given in (3.21), then the MFV predictions
are all compatible with data except for SPS4. It is also obvious from this
figure that the predictions within NMFV, as compared to the MFV ones, lie
quite generically outside the experimental allowed band, except for the above
commented deltas with low sensitivity.
2e-08
2e-08
∆ MB (MeV)
2.5e-08
∆ MB (MeV)
2.5e-08
LR
RL
δsb
δsb
δct
δsb
δ23
SM
RR
LL
RR
s
1.5e-08
s
1.5e-08
SPS1b
RL
LR
δct
δct
SPS1a
RL
LR
δct
δct
1e-08
LR
δsb
RR
δct
LL
δ23
5e-09
-0.5
0
1e-08
5e-09
SM
0.5
1
-1
-0.5
δXY
ij
SPS2
LR
δct
LR
δsb
2e-08
RR
∆ MB (MeV)
δct
LL
δ23
SM
0.5
1
0.5
1
s
1.5e-08
1e-08
1e-08
SPS3
LR
δct
δct
δsb
δsb
δct
δsb
δ23
SM
LR
5e-09
5e-09
RR
LL
-1
-0.5
0
0.5
1
-1
-0.5
δXY
ij
RL
RL
RR
0
δXY
ij
2.5e-08
2e-08
2e-08
∆ MB (MeV)
2.5e-08
∆ MB (MeV)
1
2e-08
s
1.5e-08
0.5
2.5e-08
RL
δct
RL
δsb
RR
δsb
∆ MB (MeV)
2.5e-08
0
δXY
ij
s
1.5e-08
s
1.5e-08
1e-08
SPS4
LR
δct
5e-09
-1
LR
δsb
RR
δct
LL
δ23
RL
1e-08
δct
SPS5
LR
δct
δct
δsb
δsb
δct
δsb
δ23
SM
LR
RL
δsb
RR
δsb
5e-09
RR
LL
SM
-0.5
0
0.5
1
δXY
ij
-1
-0.5
0
RL
RL
RR
δXY
ij
Figure 4. Sensitivity to the NMFV deltas in ∆MBs for the SPSX points of table 1. The experimental allowed 3σexp area is the horizontal colored band. The SM prediction and the theory
uncertainty ∆theo (∆MBs ) (red bar) is displayed on the right axis.
XY | larger than O(0.1) and still compatible with B data. In particular, δ RL , and δ RR
|δct
ct
ct
LR . The parameters on
are the less constrained parameters, and to a lesser extent also δct
the bottom-sector are, in contrast, quite constrained. The most tightly constrained are
LR and δ RL , specially the first one with just some singular allowed values: either
clearly δsb
sb
positive and of the order of 3 − 5 × 10−2 , or negative and with a small size of the order
of −7 × 10−3 ; for the second the limits are around 2 × 10−2 for both positive and negative
RR is the less constrained parameter in the bottom sector, with larger allowed
values. δsb
RR | . 0.4 − 0.9 depending on the scenario.
intervals of |δsb
– 20 –
JHEP05(2012)015
-1
RL
δsb
RR
δsb
All SPS points are defined with a positive µ value. We have checked the effect
of switching the sign of µ. While the numerical results are changing somewhat, no
qualitative change can be observed. Consequently, confining ourselves to positive µ does
not constitute a general restriction of our analysis. Similar observations are made in the
Higgs-sector analysis below.
4
Radiative corrections to MSSM Higgs masses within NMFV scenarios
In this section we present our computation of the one-loop radiative corrections to MSSM
Higgs boson masses within the NMFV scenarios. For completeness and definiteness, we
first shortly review the relevant features of the MSSM Higgs sector at tree-level. Then we
summarize the main one-loop renormalization issues that are involved in the computation
and finally we present the analytical results for the one-loop corrected Higgs masses.
4.1
The Higgs boson sector at tree-level
Contrary to the SM, in the MSSM two Higgs doublets are required. The Higgs potential [89]
V = m21 |H1 |2 + m22 |H2 |2 − m212 (ǫab H1a H2b + h.c.)
2 1
1
+ (g12 + g22 ) |H1 |2 − |H2 |2 + g22 |H1† H2 |2 ,
8
2
(4.1)
contains m1 , m2 , m12 as soft SUSY breaking parameters; g2 , g1 are the SU(2) and U(1)
gauge couplings, and ǫ12 = −1.
The doublet fields H1 and H2 are decomposed in the following way:
H1 =
H2 =
H10
!
H2+
!
H1−
H20
=
=
v1 +
v2 +
√1 (φ0 −
2 1
−φ−
1
+
φ2
1
√ (φ0 +
2 2
iχ01 )
iχ02 )
!
,
!
.
(4.2)
The potential (4.1) can be described with the help of two independent parameters (besides
g1 and g2 ): tan β = v2 /v1 and MA2 = −m212 (tan β + cot β), where MA is the mass of the
CP-odd Higgs boson A.
– 21 –
JHEP05(2012)015
The intervals allowed by B data that we have presented above will be of interest
for the following study in this work, where we will next explore the size of the radiative
corrections to the MSSM Higgs masses within these NMFV-MSSM scenarios and we will
require compatibility with B data. In the final analysis of these corrections, we will use
the constraints from B data as extracted from two non-vanishing deltas. As expected,
these constraints vary significantly respect to the ones with just one non-vanishing delta.
LL
δ23
LR
δct
RL
δct
RL
δsb
RR
δct
RR
δsb
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
Table 2. Allowed delta intervals by BR(B → Xs γ), BR(Bs → µ+ µ− ) and ∆MBs .
– 22 –
JHEP05(2012)015
LR
δsb
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
BR(B → Xs γ)
BR(Bs → µ+ µ− )
∆MBs
(-0.51:-0.43) (-0.034:0.083)
(-0.53:0.92)
(-0.73:-0.65) (-0.41:0.55) (0.73:0.79)
(-0.33:-0.27) (-0.014:0.062)
(-0.014:0.16)
(-0.090:-0.069) (-0.021:0.097) (0.14:0.17)
(-0.43:0.34) (0.90:0.92)
(-0.99:0.99)
(-0.37:0.37)
(-0.73:-0.65) (-0.083:0.12)
(-0.90:0.97)
(-0.86:-0.79) (-0.56:0.66) (0.83:0.89)
(-0.14:-0.11) (0.0069:0.034)
(0.028:0.055)
(-0.0069)(0.021:0.055)(0.076)
(-0.26:0.50)
(-0.60:0.57)
(-0.37:0.39)
(-0.89:-0.86) (-0.12:-0.097)
(-0.89:0.89)
(-0.89:0.89)
(-0.062:0.28)
(-0.44:0.67)
(-0.67:0.67)
(-0.083:0.36)
(-0.46:0.46)
(-0.46:0.46)
(-0.46:0.46)
(-0.68:0.68)
(-0.68:0.68)
(-0.43:0.61)
excluded
(-0.39:-0.021) (0.74:0.77)
(-0.61:-0.51) (0.041:0.23)
(-0.59:0.61)
(-0.59:0.61)
(-0.27:0.58)
(0)(0.034)
(-0.60:0.60)
(-0.076:0.076)
(-0.0069:0) (0.048:0.055)
(-0.43:0.54)
(-0.15:0.14)
(-0.48:0.48)
(-0.0069:0) (0.048:0.055)
(-0.19:0.19)
(-0.61:0.61)
(-0.0069:0) (0.048:0.055)
(-0.12:0.12)
(0.49)
(-0.0069)(0.034)
(-0.29:-0.24) (-0.10:-0.014) (0.12:0.18)
(-0.71:0.71)
(-0.0069:0) (0.041)
(-0.090:0.090)
(-0.84:0.84)
(-0.84:0.84)
(-0.84:0.84)
(-0.63:0.63)
(-0.63:0.63)
(-0.63:0.63)
(-0.39:0.39)
(-0.39:0.39)
(-0.39:0.39)
(-0.63:0.63)
(-0.63:0.63)
(-0.63:0.63)
excluded
excluded
(-0.72:-0.21) (0.21:0.72)
(-0.53:0.53)
(-0.53:0.53)
(-0.53:0.53)
(-0.014:0.014)
(-0.71:0.71)
(-0.069:0.069)
(-0.021:0.021)
(-0.58:0.58)
(-0.14:0.14)
(-0.014:0.014)
(-0.55:0.55)
(-0.17:0.17)
(-0.021:0.021)
(-0.63:0.63)
(-0.11:0.11)
(-0.021:-0.014)(0.014:0.021)
excluded
(-0.21:-0.17) (0.16:0.21)
(-0.014:0.014)
(-0.72:0.72)
(-0.083:0.083)
(-0.93:-0.67) (-0.64:0.93)
(-0.93:0.93)
(-0.93:0.93)
(-0.93:-0.61) (-0.56:0.90)
(-0.95:0.94)
(-0.98:0.98)
(-1.0:0.99)
(-1.0:0.99)
(-1.0:0.99)
(-0.97:0.97)
(-0.97:0.97)
(-0.98:0.97)
excluded
excluded
(-0.85:-0.22) (0.22:0.85)
(-0.60:0.60)
(-0.60:0.60)
(-0.60:0.60)
(-0.65:0.68)
(-0.96:0.96)
(-0.91:-0.90) (-0.86:-0.80) (-0.41:0.41)
(0.81:0.86) (0.90:0.91)
(-0.71:0.74)
(-0.73:0.98)
(-0.94:-0.92) (-0.83:0.88) (0.93:0.94)
(-0.99:0.99)
(-0.99:0.99)
(-0.99) (-0.39:0.39) (0.99)
(-0.98:0.98)
(-0.98:0.98)
(-0.94:-0.93) (-0.88:0.88) (0.93:0.94)
(-0.45:-0.18) (0.19:0.46)
excluded
(-0.80:-0.028) (0.461:0.71) (0.86:0.91)
(0.94:0.95)
(-0.77:0.80)
(-0.97:0.97)
(-0.92) (-0.87:-0.78) (-0.51:0.51)
(0.78:0.87) (0.92)
LL
δ23
LR
δsb
RL
δct
RL
δsb
RR
δct
RR
δsb
Total allowed intervals
(-0.034:0.083)
(-0.014:0.062)
(-0.37:0.34)
(-0.083:0.12)
(0.028:0.034)
(-0.26:0.39)
(-0.89:-0.86) (-0.12:-0.097) (-0.062:0.28)
(-0.083:0.36)
(-0.46:0.46)
(-0.43:0.61)
excluded
(-0.27:0.58)
(0)(0.034)
(-0.0069:0) (0.048:0.055)
(-0.0069:0) (0.048:0.055)
(-0.0069:0) (0.048:0.055)
excluded
(-0.0069:0) (0.041)
(-0.84:0.84)
(-0.63:0.63)
(-0.39:0.39)
(-0.63:0.63)
excluded
(-0.53:0.53)
(-0.014:0.014)
(-0.021:0.021)
(-0.014:0.014)
(-0.021:0.021)
excluded
(-0.014:0.014)
(-0.93:-0.67) (-0.64:0.93)
(-0.93:-0.61) (-0.56:0.90)
(-1.0:0.99)
(-0.97:0.97)
excluded
(-0.60:0.60)
(-0.41:0.41)
(-0.71:0.74)
(-0.99) (-0.39:0.39) (0.99)
(-0.94:-0.93) (-0.88:0.88) (0.93:0.94)
excluded
(-0.51:0.51) (0.78:0.80)
Table 3. Total allowed delta intervals by BR(B → Xs γ), BR(Bs → µ+ µ− ) and ∆MBs .
– 23 –
JHEP05(2012)015
LR
δct
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
SPS1a
SPS1b
SPS2
SPS3
SPS4
SPS5
The mixing angle α is determined through
"
#
−(MA2 + MZ2 ) sin β cos β
α = arctan
,
MZ2 cos2 β + MA2 sin2 β − m2h,tree
−
π
<α<0.
2
(4.6)
One gets the following Higgs spectrum:
2 neutral bosons, CP = +1 : h, H
1 neutral boson, CP = −1 : A
2 charged bosons : H + , H −
3 unphysical Goldstone bosons : G, G+ , G− .
(4.7)
At tree level the mass matrix of the neutral CP-even Higgs bosons is given in the
φ1 -φ2 -basis in terms of MZ , MA , and tan β by
!
2
2
m
m
2,tree
φ1
φ1 φ2
=
MHiggs
m2φ1 φ2 m2φ2
!
MA2 sin2 β + MZ2 cos2 β −(MA2 + MZ2 ) sin β cos β
=
,
(4.8)
−(MA2 + MZ2 ) sin β cos β MA2 cos2 β + MZ2 sin2 β
which by diagonalization according to eq. (4.3) yields the tree-level Higgs boson masses
!
m2H,tree
0
2,tree α
MHiggs −→
,
(4.9)
0
m2h,tree
where
(m2H,h )tree
q
1
2
2
2
2
2
2
2
2
=
MA + MZ ± (MA + MZ ) − 4MZ MA cos 2β .
2
(4.10)
The charged Higgs boson mass is given by
2
m2H ± ,tree = MA2 + MW
.
(4.11)
The masses of the gauge bosons are given in analogy to the SM:
1
2
MW
= g22 (v12 + v22 );
2
1
MZ2 = (g12 + g22 )(v12 + v22 );
2
– 24 –
Mγ = 0.
(4.12)
JHEP05(2012)015
The diagonalization of the bilinear part of the Higgs potential, i.e. of the Higgs mass
matrices, is performed via the orthogonal transformations
!
!
!
H
cos α sin α
φ01
(4.3)
=
h
− sin α cos α
φ02 ,
!
!
!
χ01
cos β sin β
G
,
(4.4)
=
χ02
− sin β cos β
A
!
!
!
φ±
cos β sin β
G±
1
.
(4.5)
=
φ±
− sin β cos β
H±
2
4.2
The Higgs boson sector at one-loop
In order to calculate one-loop corrections to the Higgs boson masses, the renormalized
Higgs boson self-energies are needed. Here we follow the procedure used in refs. [23, 90]
(and references therein) and review it for completeness. The parameters appearing in the
Higgs potential, see eq. (4.1), are renormalized as follows:
MZ2 → MZ2 + δMZ2 ,
→
→
2
MW
+
2
MHiggs
Th → Th + δTh ,
2
δMW
,
2
+ δMHiggs
,
(4.13)
TH → TH + δTH ,
tan β → tan β (1 + δtanβ ).
2
MHiggs
denotes the tree-level Higgs boson mass matrix given in eq. (4.8). Th and TH are
the tree-level tadpoles, i.e. the terms linear in h and H in the Higgs potential.
The field renormalization matrices of both Higgs multiplets can be set up symmetrically,
!
!
!
1
h
1 + 21 δZhh
h
2 δZhH
.
(4.14)
·
→
1
H
1 + 21 δZHH
H
2 δZhH
For the mass counter term matrices we use the definitions
!
2 δm2
δm
h
hH
2
δMHiggs
=
.
δm2hH δm2H
(4.15)
ˆ 2 ), can now be expressed through the unrenormalized
The renormalized self-energies, Σ(p
self-energies, Σ(p2 ), the field renormalization constants and the mass counter terms. This
reads for the CP-even part,
2
ˆ hh (p2 ) = Σhh (p2 ) + δZhh (p2 − m2
Σ
h,tree ) − δmh ,
2
2
ˆ hH (p2 ) = ΣhH (p2 ) + δZhH (p2 − 1 (m2
Σ
h,tree + mH,tree )) − δmhH ,
2
ˆ HH (p2 ) = ΣHH (p2 ) + δZHH (p2 − m2H,tree ) − δm2H .
Σ
(4.16a)
(4.16b)
(4.16c)
Inserting the renormalization transformation into the Higgs mass terms leads to
expressions for their counter terms which consequently depend on the other counter terms
introduced in (4.13).
For the CP-even part of the Higgs sectors, these counter terms are:
δm2h = δMA2 cos2 (α − β) + δMZ2 sin2 (α + β)
+
e
2MZ sw cw (δTH
(4.17a)
cos(α − β) sin2 (α − β) + δTh sin(α − β)(1 + cos2 (α − β)))
+ δtanβ sin β cos β (MA2 sin 2(α − β) + MZ2 sin 2(α + β)),
δm2hH = 21 (δMA2 sin 2(α − β) − δMZ2 sin 2(α + β))
+
e
2MZ sw cw (δTH
3
(4.17b)
3
sin (α − β) − δTh cos (α − β))
− δtanβ sin β cos β (MA2 cos 2(α − β) + MZ2 cos 2(α + β)),
δm2H = δMA2 sin2 (α − β) + δMZ2 cos2 (α + β)
– 25 –
(4.17c)
JHEP05(2012)015
2
MW
2
MHiggs
−
e
2MZ sw cw (δTH
cos(α − β)(1 + sin2 (α − β)) + δTh sin(α − β) cos2 (α − β))
− δtanβ sin β cos β (MA2 sin 2(α − β) + MZ2 sin 2(α + β)) .
For the field renormalization we chose to give each Higgs doublet one renormalization
constant,
H1 → (1 + 12 δZH1 )H1 ,
H2 → (1 + 12 δZH2 )H2 .
(4.18)
This leads to the following expressions for the various field renormalization constants
in eq. (4.14):
(4.19a)
δZhH = sin α cos α (δZH2 − δZH1 ),
(4.19b)
δZHH = cos2 α δZH1 + sin2 α δZH2 .
(4.19c)
The counter term for tan β can be expressed in terms of the vacuum expectation values as
δ tan β =
1
δv2 δv1
(δZH2 − δZH1 ) +
−
,
2
v2
v1
(4.20)
where the δvi are the renormalization constants of the vi :
v1 → (1 + δZH1 ) (v1 + δv1 ) ,
v2 → (1 + δZH2 ) (v2 + δv2 ) .
(4.21)
Similarly for the charged Higgs sector, the renormalized self-energy is expressed in terms
of the unrenormalized one and the corresponding counter-terms as:
2
ˆ H − H + p2 = ΣH − H + p2 + δZH − H + p2 − m2 ±
(4.22)
Σ
H ,tree − δmH ± ,
where,
2
δm2H ± = δMA2 + δMW
(4.23)
δZH − H + = sin2 β δZH1 + cos2 β δZH2 .
(4.24)
and,
The renormalization conditions are fixed by an appropriate renormalization scheme.
For the mass counter terms on-shell conditions are used, resulting in:
δMZ2 = Re ΣZZ (MZ2 ),
2
2
δMW
= Re ΣW W (MW
),
δMA2 = Re ΣAA (MA2 ).
(4.25)
For the gauge bosons Σ denotes the transverse part of the self-energy. Since the
tadpole coefficients are chosen to vanish in all orders, their counter terms follow from
T{h,H} + δT{h,H} = 0:
δTh = −Th ,
δTH = −TH .
(4.26)
For the remaining renormalization constants for δ tan β, δZH1 and δZH2 the most
convenient choice is a DR renormalization of δ tan β, δZH1 and δZH2 ,
idiv
h
DR
′
,
(4.27a)
=
−
Re
Σ
δZH1 = δZH
HH |α=0
1
– 26 –
JHEP05(2012)015
δZhh = sin2 α δZH1 + cos2 α δZH2 ,
idiv
h
DR
′
,
δZH2 = δZH
=
−
Re
Σ
hh |α=0
2
1
δtanβ = − (δZH2 − δZH1 ) = δtanβ DR .
2
(4.27b)
(4.27c)
Determining the poles of the matrix ∆Higgs in eq. (4.28) is equivalent to solving the equation
i2
h
ih
i h
ˆ hH (p2 ) = 0 .
ˆ hh (p2 ) p2 − m2
ˆ HH (p2 ) − Σ
p2 − m2h,tree + Σ
+
Σ
H,tree
(4.29)
Similarly, in the case of the charged Higgs sector, the corrected Higgs mass is derived by
the position of the pole in the charged Higgs propagator, which is defined by:
4.3
ˆ H − H + p2 = 0.
p2 − m2H ± ,tree + Σ
(4.30)
Analytical results of Higgs mass corrections in NMFV-SUSY
Following the previously detailed prescription for the computation of the one-loop
corrected Higgs boson masses, one finds the analytical results for these masses in terms of
the renormalized self-energies which, in turn, are written in terms of the unrenormalized
self-energies and tadpoles. To shorten the presentation of these analytical results, it is
convenient to report just on these unrenormalized self-energies and tadpoles.
The relevant one-loop corrections have been evaluated with the help of FeynArts [50–
52] and FormCalc [53]. For completeness the new Feynman rules included in the model
file are listed in the appendix A. All the results for the unrenormalized self-energies
and tadpoles are collected in appendix B. We have shown explicitly just the relevant
contributions for the present study of the radiative corrections to the Higgs boson masses
within NMFV scenarios, namely, the one-loop contributions from quarks and squarks.
The corresponding generic Feynman-diagrams for the Higgs bosons self-energies, gauge
boson self-energy diagrams and tadpole diagrams are collected in the figure 15 in appendix
B. It should also be noticed that the contributions from the squarks are the only ones
that differ from the usual ones of the MSSM with MFV. It should be noted also that the
corrections from flavor mixing, which are the subject of our interest here, are implicit in
˜
both the VCKM , and in the values of the rotation matrices, Ru˜ , Rd , and the squark masses,
mu˜i , md˜i (i = 1, . . . , 6) that appear in these formulas of the unrenormalized self-energies
and tadpoles and that have been introduced in section 2.
Finally, it is worth mentioning that we have checked the finiteness in our analytical
results for the renormalized Higgs self-energies. It is obviously expected, but it is not a
– 27 –
JHEP05(2012)015
The corresponding renormalization scale, µDR , is set to µDR = mt in all numerical
evaluations.
Finally, in the Feynman diagrammatic (FD) approach that we are following here, the
higher-order corrected CP-even Higgs boson masses are derived by finding the poles of the
(h, H)-propagator matrix. The inverse of this matrix is given by
!
2 − m2
2)
2)
ˆ
ˆ
p
+
Σ
(p
Σ
(p
HH
hH
H,tree
(∆Higgs )−1 = −i
(4.28)
ˆ hH (p2 )
ˆ hh (p2 ) .
Σ
p2 − m2h,tree + Σ
trivial check in the present scenarios with three generations of quarks and squarks and
with flavor mixing. We have also checked that the analytical results of the self-energies in
appendix B agree with the formulas in FeynHiggs [23, 24, 29–31]. Each one of the terms
contained in the appendix B was compared with the corresponding term in FeynHiggs.
During this process and the check of the finiteness, discrepancies were found with the
charged Higgs part of FeynHiggs, leading to an updated version of the code.10
5
Numerical analysis
XY
XY
∆mφ (δij
) ≡ mNMFV
(δij
) − mMSSM
,
φ
φ
φ = h, H, H ± ,
(5.1)
XY ) and mMSSM have been calculated at the one-loop level. It should be
(δij
where mNMFV
φ
φ
XY = 0) = mMSSM and, therefore, by construction, ∆m (δ XY = 0) =
noted that mNMFV
(δ
φ ij
ij
φ
φ
0, and ∆mφ gives the size of the one-loop NMFV contributions to mφ . The numerical
XY ) and mMSSM has been done with (the updated version of)
calculation of mNMFV
(δij
φ
φ
FeynHiggs [23, 24, 29–31], which solves the eqs. (4.29) and (4.30) for finding the positions
LL ) can be found in [28].
of the poles of the propagator matrix. Previous results for ∆mh (δ23
5.1
XY 6= 0
∆mφ versus one δij
We show in figures 5, 6 and 7 our numerical results for ∆mh , ∆mH and ∆mH ± ,
LL , δ LR , δ LR ,
respectively, as functions of the seven considered flavor changing deltas, δ23
ct
sb
RL , δ RL , δ RR and δ RR , which we vary in the interval −1 ≤ δ XY ≤ 1. In these plots we
δct
ct
ij
sb
sb
have chosen the same six SPS points of table 1, as for the previous study of constraints
from B physics in 2. We do not take the experimental bounds into account here, since we
just want to show the general behavior of the masses with the deltas. The experimental
bounds will be taken into account in the next subsection. As before we have checked the
impact of switching the sign of µ and found a small quantitative but no qualitative effect.
The main conclusions from these figures are the following:
- General features:
XY → −δ XY , as
All mass corrections, ∆mh , ∆mH and ∆mH ± , are symmetric δij
ij
expected. This feature is obviously different than in the previous plots of the B
observables. The sign of the mass corrections can be both positive and negative,
depending on the particular delta value. The size of the Higgs mass corrections, can
XY 6= 0 regions, reaching values even larger than 10 GeV in
be very large in some δij
10
We especially thank T. Hahn for his efforts put into this update.
– 28 –
JHEP05(2012)015
In this section we present our numerical results for the radiative corrections to the Higgs
boson masses from from flavor mixing within NMFV-SUSY scenarios. Since all one-loop
corrections in the present NMFV scenario are common to the MSSM except for the corXY values, we will focus just on the results of
rections from squarks, which depend on the δij
these corrections as a function of the flavor mixing parameters, and present the differences
with respect to the predictions within the MSSM. Correspondingly, we define:
XY | < 0.5. In
some cases, at the central region with not very large delta values, |δij
fact, the restrictions from B physics in this central region is crucial to get a reliable
estimate of these effects.
In the cases with large tan β (SPS4 and SPS1b), we also find large mass corrections
but, as already said, they are much more limited by B constraints. In particular, for
LL (see table 3).
SPS4 all deltas are excluded, except for a very narrow window in δ23
In the cases with moderate tan β = 10 (SPS1a, SPS2 and SPS3), we find large
LR , δ RL , δ LR and δ RL . The
corrections |∆mh | & 5 GeV in the central region of δsb
ct
ct
sb
other Higgs bosons get large corrections |∆mH |, |∆mH | & 5 GeV in the deltas
LR and δ RL .
central region only for δsb
sb
- Sensitivity to the various deltas:
We find very strong sensitivity in the three mass corrections ∆mh , ∆mH and ∆mH ± ,
LR and δ RL for all the seven considered SPS points.
to δsb
sb
LR and δ RL in all the conIn the case of ∆mh there is also an important sensitivity to δct
ct
sidered points. The strong sensitivity to LR and RL parameters can be understood
due to the relevance of the A-terms in these Higgs mass corrections. It can be noticed
in the Feynman rules (i.e. see the coupling of two squarks and one/two Higgs bosons
in appendix A) that the A-terms enter directly into the couplings, and in some cases,
as in the couplings of down-type squarks to the CP-odd Higgs boson, enhanced by
tan β. Therefore, considering the relationship between the A-terms and these LR and
RL parameters as is shown in eq. (2.24), the strong sensitivity to these parameters
LR in ∆m has been found in [37].
can be understood. A similar strong sensitivity to δct
h
RR . In other points,
In SPS5 there is a noticeable sensitivity to all deltas except δsb
LL
RR
the effects of δ23 , δct on ∆mh are only appreciated at the large delta region, close
RR = ±1.
to ±1. For instance, in SPS2, ∆mh = −5 GeV for δct
LR and δ RL , there is only noticeable sensitivity to
In the case of ∆mH , apart from δsb
sb
other deltas in SPS5. The same comment applies to ∆mH ± .
5.2
XY 6= 0
∆mh versus two δij
Our previous results on the Higgs mass corrections show that the corrections to the lightest
Higgs mass ∆mh are negative in many of the studied cases and can be very large for some
regions of the flavor changing deltas which are still allowed by present B data. These
– 29 –
JHEP05(2012)015
For low tan β, where the restrictions from B physics to the deltas are less severe,
the Higgs mass corrections are specially relevant. Particularly, ∆mh turns out to be
RR . For instance, at
negative and large for tan β = 5 (SPS5) for all deltas, except δsb
XY | ≃ 0.5, the mass correction ∆m for SPS5 is negative and & 5 GeV in all flavor
|δij
h
RR
changing deltas except δsb where the correction is negligible. In the case of ∆mH
and ∆mH ± the size of the correction at low tan β is smaller, . 2 GeV in the central
LR and δ RL that can also generate large corrections & 5 GeV.
region, except for δsb
sb
SPS1a
RL
LR
δct
δct
LR
δsb
RR
δct
LL
δ23
∆mh (GeV)
2
4
-2
-4
-4
0.5
1
δsb
LL
RR
δ23
-1
-0.5
δXY
ij
SPS2
LR
δct
LR
δsb
RR
δct
LL
δ23
∆mh (GeV)
2
4
RL
δct
RL
δsb
RR
δsb
δct
δsb
δsb
δct
δsb
RR
2
0
-2
-4
-4
-0.5
0
0.5
1
LL
LR
δsb
RR
δct
LL
δ23
∆mh (GeV)
2
4
RL
δct
RL
δsb
RR
δsb
-0.5
SPS5
LR
δct
δct
δsb
δsb
δct
δsb
LR
2
-2
-4
-4
0
0
0.5
1
δXY
ij
RR
LL
0.5
1
RL
RL
RR
δ23
0
-2
-0.5
1
RR
δ23
δXY
ij
0
-1
0.5
RL
-1
∆mh (GeV)
SPS4
LR
δct
1
RL
δXY
ij
4
0.5
0
-2
-1
SPS3
LR
δct
LR
∆mh (GeV)
4
0
δXY
ij
-1
-0.5
0
δXY
ij
Figure 5. Sensitivity to the NMFV deltas in ∆mh for the SPSX points of table 1.
negative and large mass corrections, can lead to a prediction for the corrected one-loop
mass in these kind of NMFV-SUSY scenarios, mNMFV
≃ mMSSM
+ ∆mh , which are indeed
h
h
too low and already excluded by present data [9, 10]. Therefore, interestingly, the study
of these mass corrections can be conclusive in the setting of additional restrictions on the
size of some flavor changing deltas which otherwise are not bounded from present B data.
In order to explore further the size of these ’dangerous’ mass corrections, we have
computed numerically the size of ∆mh as a function of two non-vanishing deltas and have
looked for areas in these two dimensional plots that are allowed by B data. We show in
figures 9, 10, 11, 12, 13, and 14 the numerical results of the ∆mh contour-lines in two
– 30 –
JHEP05(2012)015
0
δsb
δct
0
-2
-0.5
RL
δsb
RR
2
0
-1
SPS1b
RL
LR
δct
δct
LR
RL
δsb
RR
δsb
∆mh (GeV)
4
SPS1a
RL
LR
δct
δct
LR
δsb
RR
δct
LL
δ23
∆mH (GeV)
2
4
-2
-4
-4
0.5
1
δsb
LL
RR
δ23
-1
-0.5
δXY
ij
SPS2
LR
δct
LR
δsb
RR
δct
LL
δ23
∆mH (GeV)
2
4
RL
δct
RL
δsb
RR
δsb
δct
δsb
δsb
δct
δsb
RR
2
0
-2
-4
-4
-0.5
0
0.5
1
LL
∆mH (GeV)
2
LR
δsb
RR
δct
LL
δ23
4
RL
δct
RL
δsb
RR
δsb
-0.5
SPS5
LR
δct
δct
δsb
δsb
δct
δsb
LR
2
-2
-4
-4
0
0
0.5
1
δXY
ij
RR
LL
0.5
1
RL
RL
RR
δ23
0
-2
-0.5
1
RR
δ23
δXY
ij
0
-1
0.5
RL
-1
∆mH (GeV)
SPS4
LR
δct
1
RL
δXY
ij
4
0.5
0
-2
-1
SPS3
LR
δct
LR
∆mH (GeV)
4
0
δXY
ij
-1
-0.5
0
δXY
ij
Figure 6. Sensitivity to the NMFV deltas in ∆mH for the SPSX points of table 1.
LL , δ XY ), for the respective points BFP, SPS2, SPS3, SPS5, VHeavyS
dimensional plots, (δ23
ij
and HeavySLightH of table 1.
LL as one of these non-vanishing deltas mainly because
We have chosen in all plots δ23
of two reasons. First, because it is one of the most frequently studied flavor changing
parameters in the literature and, therefore, a convenient reference parameter. Second,
because there are several well motivated SUSY scenarios, where this parameter gets the
largest value, as we explained in section 2.
In these two-dimensional figures we have included the allowed/disallowed by B data
areas that have been found by following the procedure explained in section 3, and the
– 31 –
JHEP05(2012)015
0
δsb
δct
0
-2
-0.5
RL
δsb
RR
2
0
-1
SPS1b
RL
LR
δct
δct
LR
RL
δsb
RR
δsb
∆mH (GeV)
4
SPS1a
RL
LR
δct
δct
LR
δsb
RR
δct
LL
δ23
∆mH+ (GeV)
2
4
-2
-4
-4
0.5
1
δsb
LL
RR
δ23
-1
-0.5
δXY
ij
SPS2
LR
δct
LR
δsb
RR
δct
LL
δ23
∆mH+ (GeV)
2
4
RL
δct
RL
δsb
RR
δsb
δct
δsb
δsb
δct
δsb
RR
2
0
-2
-4
-4
-0.5
0
0.5
1
LL
∆mH+ (GeV)
2
LR
δsb
RR
δct
LL
δ23
4
RL
δct
RL
δsb
RR
δsb
-0.5
SPS5
LR
δct
δct
δsb
δsb
δct
δsb
LR
2
-2
-4
-4
0
0
0.5
1
δXY
ij
RR
LL
0.5
1
RL
RL
RR
δ23
0
-2
-0.5
1
RR
δ23
δXY
ij
0
-1
0.5
RL
-1
∆mH+ (GeV)
SPS4
LR
δct
1
RL
δXY
ij
4
0.5
0
-2
-1
SPS3
LR
δct
LR
∆mH+ (GeV)
4
0
δXY
ij
-1
-0.5
0
δXY
ij
Figure 7. Sensitivity to the NMFV deltas in ∆mH ± for the SPSX points of table 1.
allowed intervals are given in eqs. (3.22), (3.23) and (3.24). The color code explaining the
meaning of each colored area and the codes for the discontinuous lines are given in figure 8.
Contour lines corresponding to mass corrections above 60 GeV or below -60 GeV have not
LR,RL
been represented. In several scenarios the plots involving δsb
show a seemingly abrupt
LR,RL
behavior for |δsb
| & 0.3, corresponding to extremely large (one-loop) corrections to
mh . In general, in the case of very large one-loop corrections, in order to get a more
stable result further higher order corrections would be required, as it is known from the
higher-order corrections to mh in the MFV case (see, e.g., ref. [24]). However, we cannot
explore this possibility here. On the other hand, in order to understand the behavior of
– 32 –
JHEP05(2012)015
0
δsb
δct
0
-2
-0.5
RL
δsb
RR
2
0
-1
SPS1b
RL
LR
δct
δct
LR
RL
δsb
RR
δsb
∆mH+ (GeV)
4
LR,RL
mh as a function of δsb
a simple analytical formula would have to be extracted from
the general result. However, this is beyond the scope of our paper.
The main conclusions from these two dimensional figures are summarized in the
following:
The points that have been chosen in these plots are quite representative of all the
different patterns found. The plots for SPS1a (not shown here) manifest similar patterns
as those of SPS3. The plots for SPS1b (not shown here) manifest similar patterns as those
of BFP. The plots for SPS4 are not included because they do not manifest any allowed
areas by B data.
The largest mass corrections ∆mh found, being allowed by B data occur in plots
LL
LR ) and (δ LL , δ RL ). This applies to all the studied points. They can be as large
(δ23 , δct
ct
23
LR or δ RL close to the upper and lower horizontal bands in these
as (−50, −100) GeV at δct
ct
LR
RL
plots where δct or δct are close to ±0.5. Again these large corrections from the LR
and RL parameters are due to the A-terms, as we explained at the end of section 5.1.
Comparing the different plots, it can be seen that the size of the allowed area by the
B data (the white area inside of the colored regions) can be easily understood basically
in terms of tan β, and the heaviness of the SUSY and Higgs spectra. Generically, the
plots with largest allowed regions and with largest Higgs mass corrections correspond to
scenarios with low tan β = 5 and heavy spectra. Consequently, the cases of VHeavyS and
– 33 –
JHEP05(2012)015
Figure 8. Legend for plots of Higgs mass corrections varying two deltas simultaneously displayed
in figures 9, 10, 11, 12, 13 and 14. Each colored area represents the disallowed region by the specified
observable/s inside each box. A white area placed at the central regions of the mentioned figures
represents a region allowed by the three B observables. A white area placed outside the colored areas
represent regions of the parameter space that generate negative squared masses. These problematic
points are consequently not shown in our plots, as we did in the previous plots. The discontinuous
lines in those figures represent the contour lines for the B observables corresponding to the maximum
and minimum allowed values: dash-dot-dash for the upper bound of BR(B → Xs γ)(eq. (3.22)),
dot-dash-dot for the lower bound of BR(B → Xs γ)(eq. (3.22)), dashed line for the upper bound of
∆MBs (eq. (3.24)), a sequential three dotted line for the lower bound of ∆MBs (eq. (3.24)), and a
dotted line for the upper bound of BR(Bs → µ+ µ− )(eq. (3.23)).
JHEP05(2012)015
Figure 9. ∆mh (GeV) contour lines from our two deltas analysis for BFP. The color code for the
allowed/disallowed areas by B data is given in figure 8.
HeavySLightH are the most interesting ones, exhibiting very large radiative corrections,
resulting from the heavy SUSY spectra. In the case of HeavySLightH the large corrections
are not only found for ∆mh , but also, though to a lesser extent, for the other Higgs
bosons, ∆mH and ∆mH ± (not shown here). Consequently, in this scenario the deltas will
be very restricted by the mass bounds, especially by mh .
– 34 –
JHEP05(2012)015
Figure 10. ∆mh (GeV) contour lines from our two deltas analysis for SPS3. The color code for
the allowed/disallowed areas by B data is given in figure 8.
There are also important corrections in the allowed areas of the two dimensional plots
LL , δ RR ) for some points, particularly for SPS5 (and to a lesser extent for SPS2).
of (δ23
ct
RR
Here the corrections can be as large as -50 GeV in the upper and lower parts, i.e. for δct
close to ±0.5. In the case of SPS2 they can be up to -2 GeV for this same region.
As for the remaining two-dimensional plots they do not show relevant allowed areas
where the mass corrections are interestingly large.
– 35 –
JHEP05(2012)015
Figure 11. ∆mh (GeV) contour lines from our two deltas analysis for SPS2. The color code for
the allowed/disallowed areas by B data is given in figure 8.
6
Conclusions
In this paper we have analyzed the one-loop corrections to the Higgs boson masses in
the MSSM with Non-Minimal Flavor Violation. We assume the flavor violation is being
generated from the hypothesis of general flavor mixing in the squark mass matrices, and
XY (X, Y = L, R; i, j = t, c, u or b, s, d).
these are parametrized by a complete set of δij
– 36 –
JHEP05(2012)015
Figure 12. ∆mh (GeV) contour lines from our two deltas analysis for SPS5. The color code for
the allowed/disallowed areas by B data is given in figure 8.
In the first step of the analysis we scanned over the NMFV parameters, contrasting
them with the experimental bounds on BR(B → Xs γ), BR(Bs → µ+ µ− ) and ∆MBs . We
take into account the most up-to-date evaluations in the NMFV MSSM for BR(B → Xs γ),
BR(Bs → µ+ µ− ) and ∆MBs , as included in the BPHYSICS subroutine of the SuFla
code [42, 43].
– 37 –
JHEP05(2012)015
Figure 13. ∆mh (GeV) contour lines from our two deltas analysis for VHeavyS. The color code
for the allowed/disallowed areas by B data is given in figure 8.
For the evaluation of ∆MBs we have added the one-loop gluino boxes [44] which are
known to be very relevant in the context of NMFV scenarios [45–47]. We have estimated
the size of these corrections and compared them with the other relevant contributions
from chargino boxes and double Higgs penguins for all values of tan β for the first time.
And we have concluded that gluino boxes dominate for moderate and low tan β ≤ 20
– 38 –
JHEP05(2012)015
Figure 14. ∆mh (GeV) contour lines from our two deltas analysis for HeavySLightH. The color
code for the allowed/disallowed areas by B data is given in figure 8.
which is the interesting range for the present work. In the final part of the B physics
XY are still
analysis, we have evaluated in one-dimensional scans which intervals for the δij
allowed in certain benchmark scenarios based on the SPS points.
In the second step we analyzed the one-loop contributions of NMFV to the MSSM
Higgs boson masses, focusing on the parameter space still allowed by the experimental
– 39 –
Acknowledgments
We thank P. Paradisi and G. Isidori for kindly providing us the BPHYSICS subroutine and
for helpful discussions. We thank T. Hahn for invaluable help with FeynHiggs, as well as
FeynArts and FormCalc, and C. Pena for helpful discussions on B-physics. We are indebted
to P. Slavich for his valuable comments and corrections regarding BR(B → Xs γ). We thank
′
him for spotting the missing operators O7,8
(eqs. (3.4) and (3.5)) in a preliminary version
of this paper. S.H. thanks A. Crivellin, L. Hofer and U. Nierste for interesting discussions.
The work of S.H. was supported in part by CICYT (grant FPA 2007-66387), in part by
CICYT (grant FPA 2010-22163-C02-01) and by the Spanish MICINN’s Consolider-Ingenio
2010 Program under grant MultiDark CSD2009-00064. The work of M.H. and M.A.-C.
was partially supported by CICYT (grant FPA2009-09017) and the Comunidad de Madrid
project HEPHACOS, S2009/ESP-1473. The work of S.P. was supported by a Ram´
on y
Cajal contract from MEC (Spain) (PDRYC-2006-000930) and partially by CICYT (grant
FPA2009-09638), the Comunidad de Arag´on project DCYT-DGA E24/2 and the Generalitat de Catalunya project 2009SGR502. The work is also supported in part by the
European Community’s Marie-Curie Research Training Network under contract MRTNCT-2006-035505 ‘Tools and Precision Calculations for Physics Discoveries at Colliders’ and
also by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042).
A
Feynman rules
We list the new Feynman rules of the NMFV scenario that are involved in the present
computation. The corresponding couplings to the Higgs boson H are obtained from the
ones listened here for the lightest Higgs boson h by replacing
c α → sα
;
sα → −cα
;
sα+β → −cα+β
;
c2α → −c2α
(A.1)
The notation used for the formulas is the following: sx = sin x; cx = cos x; sw =
W
sin θW ; cw = cos θW = M
MZ ; tβ = tan β.
– 40 –
JHEP05(2012)015
flavor constraints and by current limits from Higgs boson searches. Here two relevant
XY were varied simultaneously, thus enlarging the allowed range for these parameters.
δij
We found large corrections, mainly for the low tan β case, up to several tens of GeV for
mh and somewhat smaller corrections for mH and mH ± . These corrections are specially
relevant in the case of the light MSSM Higgs boson since they can be negative and up to
two orders of magnitude larger than the anticipated LHC precision. Consequently, these
corrections must be taken into account in any Higgs boson analysis in the NMFV MSSM
framework. Conversely, in the case of a Higgs boson mass measurement these corrections
XY . The present work clearly indicates that
might be used to set further constraints on δij
LR and δ RL are severely constrained by the present bounds
the flavor mixing parameters δct
ct
on the lightest Higgs boson mass within the NMFV-MSSM scenarios.
1. Couplings of two squarks and one/two Higgs bosons
u˜i
0
h
-
P3
u
˜∗
u
˜
Ri,k
δkl Rj,l
6cα cw m2uk − MW MZ sα+β sβ (3 − 4s2w )
u
˜
+3cw Rj,3+l
(Auk,l cα muk + δkl muk µ∗ sα )
2
u
˜
u
˜∗
+Ri,3+k δkl Rj,3+l 6cα cw muk − 4MW MZ sα+β sβ s2w
u
˜
u∗
+3cw Rj,l (Al,k cα mul + δkl muk µsα )
ie
k,l=1 6MW cw sw sβ
u˜j
h0
n
P3
ie
k,l=1 6MW cw sw cβ
d˜j
n
d˜ ∗
d˜
Ri,k
δkl Rj,l
6sα cw m2dk −MW MZ sα+β cβ (3−2s2w )
o
d˜
+3cw Rj,3+l
(Adk,l sα mdk + δkl mdk µ∗ cα )
n
d˜ ∗
d˜
+Ri,3+k
δkl Rj,3+l
6sα cw m2dk −2MW MZ sα+β cβ s2w
oo
d˜
(Ad∗
+3cw Rj,l
l,k sα mdl + δkl mdk µcα )
u˜i
H−
n
d˜ P3
u∗
∗ nl
∗ kl
u
˜∗
Rj,l
Ri,3+k
n=1 An,k mun VCKM +muk µVCKM tβ
o
d˜
∗ kl
(1 + tβ 2 )
Rj,3+l
+mdl muk VCKM
n
P3
u
˜∗
d
∗ kn
∗ ∗ kl
d˜
+Ri,k
Rj,3+l
tβ
n=1 An,l mdn VCKM tβ +mdl µ VCKM
oo
∗ kl
d˜
2
+VCKM
Rj,l
m2uk − tβ (MW
s2β − m2dl tβ )
P3
ie
√
2MW sw tβ
P3
ie
√
2MW sw tβ
k,l=1
d˜j
n
d˜i
H+
k,l=1
n
u˜
P3
d˜ ∗
d∗
kn
kl
Ri,3+l
Rj,k tβ
n=1 An,l mdn VCKM tβ +mdl µVCKM
kl
u
˜
+mdl muk VCKM
Rj,3+k
(1 + tβ 2 )
u˜
P3
nl
∗ kl
d˜ ∗
u
+Ri,l
Rj,3+k
k µ VCKM tβ
n=1 An,k mun VCKM +mu
kl
u
˜
2
2
2
+VCKM Rj,k muk − tβ (MW s2β − mdl tβ )
u˜j
u˜i
0
A
-
P3
e
k,l=1 2MW sw tβ
u
˜
u
˜∗
Aul,k∗ mul + δkl muk µ tβ
Rj,l
Ri,3+k
˜
u
˜∗ u
Auk,l muk + δkl muk µ∗ tβ
Rj,3+l
−Ri,k
u˜j
d˜i
A0
-
P3
e
k,l=1 2MW sw
n
˜
˜
h
o
u˜i
-
0
˜
d∗ d
Rj,3+l Adk,l mdk tβ + δkl mdk µ∗
−Ri,k
d˜j
h0
˜
d∗
d
Ri,3+k
Rj,l
Adl,k∗ mdl tβ + δkl mdk µ
u˜j
ie2
k=1 12M 2 c2
s2 s 2
W w w β
P3
u
˜
u
˜∗
2 2
Ri,k
Rj,k
6c2α c2w m2uk − c2α MW
sβ (3 − 4s2w )
u
˜
u
˜∗
2 2 2
+2Ri,3+k
Rj,3+k
3c2α c2w m2uk − 2c2α MW
sβ sw
– 41 –
JHEP05(2012)015
d˜i
h0
d˜i
−
h0
d˜j
h0
u˜i
˜
h0
d˜i
H
ie2 s2α
k=1 12M 2 c2
s 2 s2
W w w β
P3
ie2 s2α
k=1 12M 2 c2
s2 c2
W w w β
d˜j
A0
u˜i
A
u˜j
A0
d˜i
d˜j
H−
u˜i
u˜j
H−
d˜i
d˜j
˜
o
u
˜
u
˜∗
2 2
Ri,k
Rj,k
6c2w m2uk − c2β MW
tβ (3 − 4s2w )
ie2
k=1 12M 2 c2
s2
W w w
P3
n
d˜
d˜ ∗
2
Ri,k
Rj,k
6c2w m2dk t2β + c2β MW
(3 − 2s2w )
˜
˜
d
d∗
2 2
+2Ri,3+k
Rj,3+k
3c2w m2dk t2β + c2β MW
sw
ie2
k,l=1 12M 2 s2
c2 t2
W w w β
P3
u
˜∗ u
˜ 2
Ri,k
Rj,l
tβ 6
P3
n=1
o
∗ kn
ln
m2dn VCKM
VCKM
c2w t2β
2
+c2β δkl MW
(1 + 2c2w )
2 2 2
u
˜
u
˜∗
3c2w m2uk − 2c2β MW
s w tβ
+2δkl Ri,3+k Rj,3+l
−
H+
2 2 2
u
˜∗
u
˜
3c2w m2uk − 2c2β MW
tβ s w
Rj,3+k
+2Ri,3+k
−
H+
ie2
k=1 12M 2 c2
s2 t 2
W w w β
P3
o
n
2 2
d˜ ∗
d˜
cβ (3 − 2s2w )
3c2w m2dk − MW
Rj,k
Ri,k
˜
−
A0
u
˜
u
˜∗
2 2
Ri,k
Rj,k
3c2w m2uk − MW
sβ (3 − 4s2w )
d
d∗
2 2 2
+Ri,3+k
Rj,3+k
3c2w m2dk − 2MW
cβ sw
−
0
u
˜
u
˜∗
2 2 2
+Ri,3+k
Rj,3+k
3c2w m2uk − 4MW
sβ sw
P3
H0
˜
JHEP05(2012)015
u˜j
n
2 2
d˜ ∗
d˜
cβ (3 − 2s2w )
6s2α c2w m2dk + c2α MW
Rj,k
Ri,k
ie2
k=1 12M 2 c2
s2 c2
W w w β
d
d∗
2 2 2
+2Ri,3+k
Rj,3+k
3s2α c2w m2dk + c2α MW
cβ sw
0
P3
ie2
k,l=1 12M 2 s2
c2 t2
W w w β
P3
n
P
d˜ ∗ d˜
nk
∗ nl 2
Ri,k
Rj,l 6 3n=1 m2un VCKM
VCKM
cw
2 2
+c2β δkl MW
tβ (1 − 4c2w )
˜
˜
d∗
d
2 2
+2δkl Ri,3+k
Rj,3+l
t2β 3c2w t2β m2dk + c2β MW
sw
– 42 –
o
3. Couplings of two squarks and one/two gauge bosons
u˜i
Zµ
P3
ie
k=1 6cw sw
′
u
˜∗
u
˜
u
˜∗ u
˜
4Ri,3+k
Rj,3+k
s2w − Ri,k
Rj,k
(3 − 4s2w ) (p + p )µ
u˜j
d˜i
′
d˜ ∗
d˜
d˜ ∗ d˜
2Ri,3+k
Rj,3+k
s2w − Ri,k
Rj,k (3 − 2s2w ) (p + p )µ
−
P3
−
P3
∗ kl
u
˜ ∗ d˜
√ie VCKM
Ri,k
Rj,l
2sw
(p + p )µ
−
P3
kl
u
˜
d˜∗
√ie VCKM
Rj,k
Ri,l
2sw
(p + p )µ
ie
k=1 6cw sw
d˜j
Wµ−
u˜i
k,l=1
′
d˜j
Wµ+
d˜i
k,l=1
′
u˜j
Zµ
u˜i
ie2
2
k=1 18c2
w sw
P3
Zν
u˜j
Zµ
d˜i
ie2
2
k=1 18c2
w sw
P3
Zν
d˜j
Wµ−
u˜i
u
˜∗ u
˜
u
˜∗
u
˜
Ri,k
Rj,k
(3 − 4s2w )2 + 16Ri,3+k
Rj,3+k
s4w gµν
d˜ ∗ d˜
d˜ ∗
d˜
Ri,k
Rj,k (3 − 2s2w )2 + 4Ri,3+k
Rj,3+k
s4w gµν
u
˜∗ u
˜
ie2
Ri,k
Rj,k
k=1 2s2
w
gµν
d˜∗ d˜
ie2
Ri,k
Rj,k
k=1 2s2
w
gµν
P3
Wν+
u˜j
Wµ−
d˜i
P3
Wν+
d˜j
– 43 –
JHEP05(2012)015
Zµ
q˜
q
q˜
φ’
φ’
φ
φ
φ
q’
q˜ ’
q
q˜
q˜
V
V
V
V
V
V
q
q˜
φ
Figure 15. Different topologies for Σφφ′ , ΣV V , Tφ .
B
Tadpoles and self-energies
All the following Feynman diagrams have been calculated using FeynArts 3.5 [50–52] and
FormCalc 6.0 [53]. The notation used here is the same as in appendix A. Furthermore we
use the functions [92]
Z 4−D D
µ
d k
1
i
A0 m2 ≡
(B.1)
D
2
16π
k − m2
(2π)
Z 4−D D
i
µ
d k
1
i
B0 p2 , m21 , m22 ≡
(B.2)
h
D 2
16π
2
(2π)
k − m (k + p)2 − m2
1
i 2 2 2 2
p B1 p , m1 , m2 ≡
16π
Z
µ4−D dD k
(2π)
D
2
pk
h
i
k 2 − m21 (k + p)2 − m22
(B.3)
The generic diagrams have been ordered according to its topologies, and the particles
involved in the internal loops (quarks q or squarks q˜). The diagrams can be found in
figure 15. The complete self-energy can be expressed as a sum of three parts:
1˜
q
2˜
q
Σφφ′ = Σ2q
φφ′ + Σφφ′ + Σφφ′
1˜
q
2˜
q
ΣV V = Σ2q
V V + ΣV V + ΣV V
Tφ = Tφq + Tφq˜
(B.4)
where φ, φ′ = h, H, A, H ± and V = W, Z. All the self-energies Σ correspond to Σ p2 .
The self-energies for H are obtained by the replacements of eq. (A.1) on the results of h:
• h
Σ2q
hh
=−
−
3
X
i=1
3
X
3αc2α m2ui 2 A0 mui + p2 B1 p2 , m2ui , m2ui + 2m2ui B0 p2 , m2ui , m2ui
2
2
2
4πMW sβ sW
3αs2α m2di
2 c 2 s2
4πMW
β W
i=1
A0 m2di + p2 B1 p2 , m2di , m2di + 2m2di B0 p2 , m2di , m2di (B.5)
– 44 –
JHEP05(2012)015
q˜ ’
q’
φ
φ’
q
Σ2˜
hh =
6 X
3
X
1
m,n i,j,k,l
2 πs2 s2
48c2W MW
β W
h
i
αB0 p2 , mu2˜m , m2u˜n
(
u˜ u˜∗
Rm,i
× δi,j MW mZ sα+β sβ −3 + 4s2W + 6cα cW m2ui Rn,j
u
˜
u
˜∗
Rm,i
+3cW cα Aui,j + µ∗ sα δi,j mui Rn,3+j
u
˜
u
˜∗
+2δi,j −2MW mZ sα+β sβ s2W + 3cα cW m2ui Rn,3+j
Rm,3+i
(
)
JHEP05(2012)015
u˜∗
u
˜
u
˜
+ 3cα cW Au∗
j,i muj Rn,j + 3cW µsα δi,j mui Rn,j Rm,3+i
u˜
u
˜∗
× δk,l MW mZ sα+β sβ −3 + 4s2W + 6cα cW m2uk Rm,l
Rn,k
u
˜
u
˜∗
+3cW cα Auk,l + µ∗ sα δk,l muk Rm,3+l
Rn,k
u˜∗
u
˜
u
˜
+ 3cα cW Au∗
l,k mul Rm,l + 3cW µsα δk,l muk Rm,l Rn,3+k
u
˜
u
˜∗
+2δk,l −2MW mZ sα+β sβ s2W + 3cα cW m2uk Rm,3+l
Rn,3+k
+
6 X
3
X
m,n i,j,k,l
1
2 πc2 s2
48c2W MW
β W
)
h
i
αB0 p2 , m2d˜ , m2d˜
m
n
(
d˜ d∗
˜
Rm,i
× δi,j MW mZ sα+β cβ −3 + 2s2W + 6sα cW m2di Rn,j
˜
d˜
d∗
Rm,i
+3cW sα Adi,j + µ∗ cα δi,j mdi Rn,3+j
˜
d˜
d˜
d∗
+ 3sα cW Ad∗
m
R
+
3c
µc
δ
m
R
α i,j di n,j Rm,3+i
W
j,i dj n,j
−2δi,j MW mZ sα+β cβ s2W − 3sα cW m2di
×
(
˜
d˜
d∗
Rn,3+j
Rm,3+i
d˜ d∗
˜
δk,l MW mZ sα+β cβ −3 + 2s2W + 6sα cW m2dk Rm,l
Rn,k
˜
d˜
d∗
+3cW sα Adk,l + µ∗ cα δk,l mdk Rm,3+l
Rn,k
˜
d˜
d˜
d∗
+ 3sα cW Ad∗
m
R
+
3c
µc
δ
m
R
α k,l dk m,l Rn,3+k
W
l,k dl m,l
−2δk,l MW mZ sα+β cβ s2W − 3sα cW m2dk
q
Σ1˜
hh =
)
3
6 X
X
l=1 i=1
1
2 πs2 s2
16c2W MW
β W
˜
d∗
d˜
Rn,3+k
Rm,3+l
)
(B.6)
h
in
u
˜ u
˜∗
c2α m2w s2β −3 + 4s2W + 6c2α c2W m2ui
αA0 mu2˜l Rl,i
Rl,i
2 2 2
u
˜∗
u
˜
−2c2α MW
sβ sW + 3c2α c2W m2ui
Rl,3+i
+2Rl,3+i
– 45 –
o
6 X
3
X
−
1
h
in
˜
d˜ d∗
αA0 m2d˜ Rl,i
Rl,i c2α m2w c2β −3+2s2W −6s2α c2W m2di
2 πc2 s2
16c2W MW
β W
l=1 i=1
˜
2 2 2
d˜
d∗
c2α MW
c β sW
−2Rl,3+i
Rl,3+i
l
+ 3s2α c2W m2di
• hH
o
(B.7)
3
X
3αcα sα m2di 2 2 2 2
A0 mdi +p B1 p , mdi , m2di +2m2di B0 p2 , m2di , m2di
+
2
2
2
4πMW cβ sW
i=1
q
Σ2˜
hH =
6 X
3
X
m,n i,j,k,l
1
2 πs2 s2
48c2W MW
β W
h
i
αB0 p2 , m2u˜m , mu2˜n
(
u˜ u˜∗
Rm,i
× δi,j MW mZ sα+β sβ −3 + 4s2W + 6cα cW m2ui Rn,j
u
˜
u
˜∗
Rm,i
+3cW cα Aui,j + µ∗ sα δi,j mui Rn,3+j
u˜∗
u
˜
u
˜
+ 3cα cW Au∗
j,i muj Rn,j + 3cW µsα δi,j mui Rn,j Rm,3+i
u
˜
u
˜∗
Rm,3+i
+2δi,j −2MW mZ sα+β sβ s2W + 3cα cW m2ui Rn,3+j
(
)
u˜
u
˜∗
Rn,k
× δk,l MW mZ cα+β sβ 3 − 4s2W + 6sα cW m2uk Rm,l
u
˜
u
˜∗
+3cW sα Auk,l − µ∗ cα δk,l muk Rm,3+l
Rn,k
u˜∗
u
˜
u
˜
+ 3sα cW Au∗
l,k mul Rm,l − 3cW µcα δk,l muk Rm,l Rn,3+k
u
˜
u
˜∗
+2δk,l 2MW mZ cα+β sβ s2W + 3sα cW m2uk Rm,3+l
Rn,3+k
−
3
6 X
X
m,n i,j,k,l
1
2 πc2 s2
48c2W MW
β W
)
h
i
αB0 p2 , m2d˜ , m2d˜
m
n
(
d˜ d∗
˜
× δi,j MW mZ sα+β cβ −3 + 2s2W + 6sα cW m2di Rn,j
Rm,i
˜
d˜
d∗
Rm,i
+3cW sα Adi,j + µ∗ cα δi,j mdi Rn,3+j
˜
d∗
d˜
d˜
+
3c
µc
δ
m
R
+ 3sα cW Ad∗
m
R
α i,j di n,j Rm,3+i
W
j,i dj n,j
−2δi,j MW mZ sα+β cβ s2W − 3sα cW m2di
(
˜
d˜
d∗
Rn,3+j
Rm,3+i
)
d˜ d∗
˜
Rn,k
× δk,l MW mZ cα+β cβ −3 + 2s2W + 6cα cW m2dk Rm,l
– 46 –
(B.8)
JHEP05(2012)015
3
X
3αcα sα m2ui 2 2 2 2
A0 mui +p B1 p , mui , m2ui +2m2ui B0 p2 , m2ui , m2ui
Σ2q
=
−
hH
2
2
2
4πMW sβ sW
i=1
˜
d∗
d˜
+3cW cα Adk,l − µ∗ sα δk,l mdk Rm,3+l
Rn,k
˜
d˜
d˜
d∗
+ 3cα cW Ad∗
l,k mdl Rm,l − 3cW µsα δk,l mdk Rm,l Rn,3+k
−2δk,l 2MW mZ cα+β cβ s2W
q
Σ1˜
hH =
3
6 X
X
l=1 i=1
1
2 πs2 s2
16c2W MW
β W
−
3cα cW m2dk
˜
d˜
d∗
Rm,3+l
Rn,3+k
h
in
˜
d˜ d∗
αA0 m2d˜ Rl,i
Rl,i s2α m2w c2β −3+4s2W +3s2α c2W m2di
2 πc2 s2
16c2W MW
β W
l=1 i=1
˜
˜
2 2 2
d∗
d
−2s2α MW
c β sW
Rl,3+i
+Rl,3+i
l
+ 3s2α c2W m2di
• A
Σ2q
AA
=−
−
q
Σ2˜
AA
3
X
i=1
3
X
o
o
(B.10)
2 2
2
3αm2ui
2
2
m
p
,
m
,
m
A
+
p
B
0
1
u
u
u
i
i
i
2 t2 s 2
4πMW
β W
3αt2β m2di 2 s2
4πMW
W
i=1
=−
6 X
3
X
−
3
6 X
X
A0 m2di + p2 B1 p2 , m2di , m2di
(B.11)
h
i
3
2
2
2
p
,
m
,
m
αB
0
u
˜
u
˜
m
n
2 πt2 s2
16MW
β W
m,n i,j,k,l
n
u˜ u˜∗ o
u
˜
u
˜∗
Rm,i
+ Au∗
× −Aui,j − µ∗ tβ δi,j mui Rn,3+j
j,i muj + µtβ δi,j mui Rn,j Rm,3+i
n
o
u˜
u
˜
u
˜∗
u
˜∗
Rn,k
+ Au∗
× −Auk,l − µ∗ tβ δk,l muk Rm,3+l
l,k mul + µtβ δk,l muk Rm,l Rn,3+k
3
i
h
αB0 p2 , m2d˜ , m2d˜
2 πs2
m
n
16MW
W
o
n
˜
˜
d˜
d∗
d˜
d∗
R
R
Rm,i
+ tβ Ad∗
m
+
µδ
m
× −tβ Adi,j − µ∗ δi,j mdi Rn,3+j
i,j di
n,j m,3+i
j,i dj
n
o
˜
˜
d˜
d∗
d˜
d∗
× −tβ Adk,l −µ∗ δk,l mdk Rm,3+l
Rn,k
+ tβ Ad∗
m
+µδ
m
R
R
(B.12)
k,l dk
l,k dl
m,l n,3+k
q
Σ1˜
AA
=
m,n i,j,k,l
3
6 X
X
h
in
1
u
˜ u
˜∗
c2β m2w t2β −3 + 4s2W + 6c2W m2ui
m
Rl,i
Rl,i
αA
u
˜
0
l
2
2
2
2
16cW MW πtβ sW
l=1 i=1
o
u
˜
u
˜∗
2 2 2
+2Rl,3+i
Rl,3+i
−2c2β MW
tβ sW + 3c2W m2ui
3
6 X
X
h
in
1
˜
d˜ d∗
m
αA
Rl,i
Rl,i c2β m2w 3 − 2s2W + 6c2W t2β m2di
0
d˜l
2
2
2
16cW MW πsW
l=1 i=1
o
˜
2 2
d˜
d∗
c2β MW
sW + 3c2W t2β m2di
(B.13)
+2Rl,3+i
Rl,3+i
+
– 47 –
JHEP05(2012)015
1
(B.9)
h
in
u
˜ u
˜∗
s2α m2w s2β −3+4s2W +3s2α c2W m2ui
αA0 m2u˜l Rl,i
Rl,i
2 2 2
u
˜
u
˜∗
−4s2α MW
sβ sW + 3s2α c2W m2ui
+Rl,3+i
Rl,3+i
3
6 X
X
−
)
• H±
Σ2q
H−H+
3 X
3
X
=−
(
h
i
3α
i,j∗
i,j
2
2
2
2
2 2
2
2
2
m
2m
+m
/t
+m
t
B
V
V
p
,
m
,
m
ui
ui β
ui
dj
dj β
dj
CKM CKM 0
2 s2
4πMW
W
i=1 j=1
h
i
i,j
i,j∗ 2
VCKM
p B1 p2 , m2ui , m2dj
+ m2ui /t2β + m2dj t2β VCKM
)
h
i
i,j∗ 2
i,j
i,j∗
i,j
tβ
VCKM
(B.14)
/t2β + m2dj VCKM
VCKM
+A0 m2dj m2ui VCKM
=−
×
3
6 X
X
m,n i,j,k,l
(
2 2
3
2
p
,
m
,
m
αB
0
u
˜
u
˜
m
n
2 πt2 s2
8MW
β W
"
3
X
k,p∗
p,i∗
d˜
u
˜∗
d˜
u
˜∗
t2β Adp,i VCKM
mdp Rn,3+i
Rm,k
+ Au∗
p,k VCKM mup Rn,i Rm,3+k
p,q
˜
˜
q,j
l,q
u
˜
d∗
u
u
˜
d∗
R
R
+
A
V
m
R
R
× t2β Ad∗
V
m
q,j CKM dq m,l n,3+j
q,l CKM uq m,3+l n,j
+
3
X
p
"
#
n
k,i∗
u
˜∗
u
˜∗
d˜
2
+ µtβ muk Rm,3+k
Rn,i
−MW
s2β tβ + m2uk + t2β m2di Rm,k
VCKM
o
u
˜∗
u
˜∗
d˜
µ∗ tβ Rm,k
+ 1 + t2β muk Rm,3+k
+mdi Rn,3+i
o
n
˜
˜
l,p
p,j
d∗
u
˜
d∗
u
˜
+ t2β Ad∗
mup Rm,3+l
Rn,j
× Aup,l VCKM
p,j VCKM mdp Rm,l Rn,3+j
n
˜
˜
l,j
u
˜
d∗
d∗
2
Rm,l
+ VCKM
+ µtβ mdj Rn,3+j
−MW
s2β tβ + m2ul + t2β m2dj Rn,j
o
˜
˜
d∗
d∗
u
˜
µ∗ tβ Rn,j
+ 1 + t2β mdj Rn,3+j
+mul Rm,3+l
#
n
o
k,p∗
d˜
u
˜∗
d˜
u
˜∗
2 d
u∗ p,i
× Ap,k VCKM mup Rn,i Rm,3+k + tβ Ap,i VCKM mdp Rn,3+i Rm,k
"
n
l,j
k,i∗
d˜
Rn,i
+ VCKM
VCKM
u˜∗
2
u
˜∗
−MW
s2β tβ + m2uk + t2β m2di Rm,k
+ µtβ muk Rm,3+k
o
d˜
u
˜∗
u
˜∗
+mdi Rn,3+i
µ∗ tβ Rm,k
+ 1 + t2β muk Rm,3+k
n
˜
˜
u
˜
2
2
d∗
2 2
d∗
× Rm,l −MW s2β tβ + mul + tβ mdj Rn,j + µtβ mdj Rn,3+j
#)
o
˜
˜
u
˜
d∗
∗
d∗
2
+mul Rm,3+l µ tβ Rn,j + 1 + tβ mdj Rn,3+j
q
Σ1˜
H−H+
3
6 X
X
(B.15)
(
h
i
1
2 2 2
u
˜
u
˜∗
−2c2β MW
tβ sW + 3c2W m2ui
2Rl,3+i
Rl,3+i
αA0 m2u˜l
=
2
2
2
2
16cW MW πtβ sW
l=1 i=1


3
3 X

X
j,k∗
u
˜∗ 2 2 i,k
u
˜  u
˜∗
6Rl,j
cW tβ VCKM VCKM
m2dk 
+t2β Rl,i
Rl,i
c2β m2w 1 + 2c2W +

j=1 k=1
– 48 –
JHEP05(2012)015
q
Σ2˜
H−H+
6 X
3
X
(
h
i
1
˜
2 2 2
2 4 2
2
d∗
d˜
c
M
t
s
+
3c
t
m
m
R
αA
2R
+
0
2β
˜
W
W
W
β
β
d
l,3+i
l,3+i
2 πt2 s2
i
dl
16c2W MW
β W
l=1 i=1


3
3 X

X
˜ 2
˜
k,i∗
k,j
d∗
2
d˜  d∗
2 2
m2uk 
VCKM
6Rl,j
cW VCKM
(B.16)
+Rl,i Rl,i c2β mw tβ 1 − 4cW +

j=1 k=1
• Z
Σ2q
ZZ
q
Σ2˜
ZZ = −
6 X
3
X
m,n i,j
n h
i
h
i
α
2
2
2
2
2
m
A
+
2m
p
,
m
,
m
B
0
u
˜n
u
˜m 0
u
˜m
u
˜n
72c2W πs2W
h
i p2
2
2
2
2
2
+ mu˜m − mu˜n
+ p +
−
B1 p , mu˜m , mu˜n −
3
u˜ u˜∗
u
˜
u
˜∗
Rm,i + 4s2W Rn,3+i
Rm,3+i
× −3 + 4s2W Rn,i
u˜
u
˜∗
u
˜
u
˜∗
−3 + 4s2W Rm,j
Rn,j
+ 4s2W Rm,3+j
Rn,3+j
3
6 X
i
h
i
n h
X
α
2
2
2
2
2
−
p
,
m
,
m
+
2m
B
m
A
0
0
˜
˜
˜
˜
dm
dn
dm
dn
72c2W πs2W
m,n i,j
h
i p2
2
2
2
2
2
2
2
2
+ md˜ − md˜
+ p + md˜ − md˜ B1 p , md˜ , md˜ −
m
n
m
n
m
n
3
o
n
˜
˜
d˜
d∗
d˜
d∗
Rm,i
+ 2s2W Rn,3+i
Rm,3+i
× −3 + 2s2W Rn,i
o
n
d˜
˜
˜
d∗
d˜
d∗
Rn,3+j
+ 2s2W Rm,3+j
−3 + 2s2W Rm,j
Rn,j
2
q
Σ1˜
ZZ
=
6 X
3
X
mu2˜m
+
l=1 i=1
Σ2q
WW = −
3
X
i,l
m2u˜n
(B.18)
h
i
1
2 2
4
u
˜
u
˜∗
2
u
˜ u
˜∗
−3
+
4s
+
16s
R
R
m
αA
R
R
0
W
W
u
˜
l,3+i l,3+i
l,i l,i
l
24c2W πs2W
l=1 i=1
3
6 X
X
• W
(B.17)
h
i
1
˜
˜
2
2 2
4
d˜
d∗
d˜ d∗
m
αA
3−2s
+4s
R
R
R
R
(B.19)
0
W
W l,3+i l,3+i
l,i l,i
d˜l
24c2W πs2W
α
l,i
l,i∗ 2A0 m2di + m2ul B0 p2 , m2di , m2ul
VCKM
VCKM
2
4πsW
– 49 –
JHEP05(2012)015
3
X
9 − 24s2W + 32s4W α 2 =−
A0 mui + p2 B1 p2 , m2ui , m2ui
2
2
36cW πsW
i=1
3
X
2 2
9 + 48s2W − 64s4W α 2
2
−
p
,
m
,
m
m
B
0
u
u
u
i
i
i
72c2W πs2W
i=1
3
X
9 − 12s2W + 8s4W α 2 −
A0 mdi + p2 B1 p2 , m2di , m2di
2
2
36cW πsW
i=1
3
X
2 2
9 + 24s2W − 16s4W α 2
2
p
,
m
,
m
−
m
B
0
d
d
d
i
i
i
72c2W πs2W
i=1
+ m2dl − m2ul + 2p2 B1 p2 , m2ui , m2dl
(B.20)
3
6 X
X
i
n h
3α
˜
l,j∗
k,i
u
˜
u
˜∗
d˜
d∗
2
R
R
R
R
m
V
V
A
0
˜
n,j
n,i
m,k
m,l
dn
12πs2W CKM CKM
m,n i,j,k,l
i
h
+2mu2˜m B0 p2 , m2u˜m , m2d˜
n
h
i p2
2
2
2
2
2
2
2
2
+ p + mu˜m − md˜ B1 p , mu˜m , md˜ −
+ mu˜m + md˜
n
n
n
3
q
Σ2˜
WW
= −
3
6 X
X
=
l=1 i=1
3
6
h
h
i
i
XX 3
3
˜
2
2
u
˜ u
˜∗
d˜ d∗
m
m
αA
αA
R
R
+
Rl,i
Rl,i (B.22)
0
0
u
˜l
l,i l,i
2
d˜l
8πs2W
8πs
W
i=1
l=1
• Tadpoles
Thq = −
Thq˜ =
3
X
3
8π 2 M
i
6 X
3
X
m
×
i,j
"(
W sβ sW
32π 2 c
cα m2ui eA0 m2ui
(B.23)
1
eA0 mu2˜m
W MW s β s W
u˜
δi,j MW mZ sα+β sβ −3 + 4s2W + 6cα cW m2ui Rm,j
+3cW
cα Aui,j
+µ
∗
u
˜
sα δi,j mui Rm,3+j
(
)
u
˜∗
Rm,i
u
˜
u
˜
+ 3cα cW Au∗
j,i muj Rm,j + 3cW µsα δi,j mui Rm,j
+2δi,j −2MW mZ sα+β sβ s2W
+
3
X
3
8π 2 MW cβ sW
i
×
"(
u
˜
Rm,3+j
3
6 X
X
sα m2di eA0 m2di −
m
i,j
)
u
˜∗
Rm,3+i
#
h
i
1
2
m
eA
0
d˜m
32π 2 cW MW cβ sW
d˜
δi,j MW mZ sα+β cβ −3 + 2s2W + 6sα cW m2di Rm,j
+3cW
(
+
3cα cW m2ui
sα Adi,j
∗
+ µ cα δi,j
d˜
mdi Rm,3+j
)
˜
d∗
Rm,i
˜
˜
d
d
+ 3sα cW Ad∗
j,i mdj Rm,j + 3cW µcα δi,j mdi Rm,j
−2δi,j MW mZ sα+β cβ s2W
−
3sα cW m2di
– 50 –
d˜
Rm,3+j
)
˜
d∗
Rm,3+i
#
(B.24)
JHEP05(2012)015
q
Σ1˜
WW
(B.21)
The self-energies for TH are obtained using the replacements of eq. (A.1) on the
results of Th .
References
[1] H.P. Nilles, Supersymmetry, supergravity and particle physics, Phys. Rept. 110 (1984) 1
[INSPIRE].
[2] H.E. Haber and G.L. Kane, The search for supersymmetry: probing physics beyond the
Standard Model, Phys. Rept. 117 (1985) 75 [INSPIRE].
[4] S. Heinemeyer, MSSM Higgs physics at higher orders, Int. J. Mod. Phys. A 21 (2006) 2659
[hep-ph/0407244] [INSPIRE].
[5] A. Djouadi, The anatomy of electro-weak symmetry breaking. II. The Higgs bosons in the
minimal supersymmetric model, Phys. Rept. 459 (2008) 1 [hep-ph/0503173] [INSPIRE].
[6] W. Ehrenfeld, SUSY searches at ATLAS, talk given at SUSY11, https://indico.fnal.gov/
contributionDisplay.py?sessionId=10&contribId=257&confId=3563, Fermilab, Batavia
U.S.A. August 2011.
[7] I. Melzer-Pellmann, SUSY searches at CMS, talk given at SUSY11, https://indico.fnal.gov/
contributionDisplay.py?sessionId=10&contribId=258&confId=3563, Fermilab, Batavia
U.S.A. August 2011.
[8] Particle Data Group collaboration, K. Nakamura et al., Review of particle physics,
J. Phys. G 37 (2010) 075021 [INSPIRE].
[9] LEP Working Group for Higgs boson searches, ALEPH, DELPHI, L3 and OPAL
collaborations, R. Barate et al., Search for the Standard Model Higgs boson at LEP,
Phys. Lett. B 565 (2003) 61 [hep-ex/0306033] [INSPIRE].
[10] ALEPH, DELPHI, L3, OPAL and LEP Working Group for Higgs Boson Searches
collaborations, S. Schael et al., Search for neutral MSSM Higgs bosons at LEP,
Eur. Phys. J. C 47 (2006) 547 [hep-ex/0602042] [INSPIRE].
[11] Tevatron new phenomena & Higgs working group webpage, http://tevnphwg.fnal.gov/.
[12] M. Vazquez Acosta, Higgs searches at the LHC, talk given at SUSY11, https://indico.
fnal.gov/contributionDisplay.py?sessionId=13&contribId=261&confId=3563, Fermilab,
Batavia U.S.A. August 2011.
[13] ATLAS collaboration, G. Aad et al., Expected performance of the ATLAS experiment —
detector, trigger and physics, arXiv:0901.0512 [INSPIRE].
[14] CMS collaboration, G. Bayatian et al., CMS technical design report, volume II: Physics
performance, J. Phys. G 34 (2007) 995 [INSPIRE].
[15] V. B¨
uscher and K. Jakobs, Higgs boson searches at hadron colliders,
Int. J. Mod. Phys. A 20 (2005) 2523 [hep-ph/0504099] [INSPIRE].
[16] ECFA/DESY LC Physics Working Group collaboration, J. Aguilar-Saavedra et al.,
TESLA: the superconducting electron positron linear collider with an integrated x-ray laser
laboratory. TDR part 3. Physics at an e+ e− linear collider, hep-ph/0106315 [INSPIRE].
– 51 –
JHEP05(2012)015
[3] R. Barbieri, Looking beyond the Standard Model: the supersymmetric option,
Riv. Nuovo Cim. 11 (1988) 1.
[17] TESLA — the superconducting electron-positron linear collider with an integrated X-ray laser
laboratory technical design report webpage,
http://tesla.desy.de/new pages/TDR CD/start.html.
[18] K. Ackermann et al., Extended joint ECFA/DESY study on physics and detectors for a
linear electron-positron collider, prepared for 4th ECFA/DESY Workshop on Physics and
Detectors for a 90 GeV to 800 GeV Linear e+ e− Collider, DESY-PROC-2004-01, Amsterdam
The Netherlands April 1–4 2003 [INSPIRE].
[19] S. Heinemeyer et al., Toward high precision Higgs-boson measurements at the international
linear e+ e− collider, hep-ph/0511332 [INSPIRE].
[21] A. De Roeck et al., From the LHC to future colliders, Eur. Phys. J. C 66 (2010) 525
[arXiv:0909.3240] [INSPIRE].
[22] S. Heinemeyer, W. Hollik, H. Rzehak and G. Weiglein, The Higgs sector of the complex
MSSM at two-loop order: QCD contributions, Phys. Lett. B 652 (2007) 300
[arXiv:0705.0746] [INSPIRE].
[23] M. Frank et al., The Higgs boson masses and mixings of the complex MSSM in the
Feynman-diagrammatic approach, JHEP 02 (2007) 047 [hep-ph/0611326] [INSPIRE].
[24] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich and G. Weiglein, Towards high precision
predictions for the MSSM Higgs sector, Eur. Phys. J. C 28 (2003) 133 [hep-ph/0212020]
[INSPIRE].
[25] S.P. Martin, Three-loop corrections to the lightest Higgs scalar boson mass in supersymmetry,
Phys. Rev. D 75 (2007) 055005 [hep-ph/0701051] [INSPIRE].
[26] R.V. Harlander, P. Kant, L. Mihaila and M. Steinhauser, Higgs boson mass in
supersymmetry to three loops, Phys. Rev. Lett. 100 (2008) 191602
[Erratum ibid. 101 (2008) 039901] [arXiv:0803.0672] [INSPIRE].
[27] P. Kant, R. Harlander, L. Mihaila and M. Steinhauser, Light MSSM Higgs boson mass to
three-loop accuracy, JHEP 08 (2010) 104 [arXiv:1005.5709] [INSPIRE].
[28] S. Heinemeyer, W. Hollik, F. Merz and S. Pe˜
naranda, Electroweak precision observables in
the MSSM with Non-Minimal Flavor Violation, Eur. Phys. J. C 37 (2004) 481
[hep-ph/0403228] [INSPIRE].
[29] S. Heinemeyer, W. Hollik and G. Weiglein, FeynHiggs: a program for the calculation of the
masses of the neutral CP even Higgs bosons in the MSSM,
Comput. Phys. Commun. 124 (2000) 76 [hep-ph/9812320] [INSPIRE].
[30] FeynHiggs webpage, http://www.feynhiggs.de/.
[31] S. Heinemeyer, W. Hollik and G. Weiglein, The masses of the neutral CP-even Higgs bosons
in the MSSM: accurate analysis at the two loop level, Eur. Phys. J. C 9 (1999) 343
[hep-ph/9812472] [INSPIRE].
[32] J. Guasch and J. Sol`
a, FCNC top quark decays: a door to SUSY physics in high luminosity
colliders?, Nucl. Phys. B 562 (1999) 3 [hep-ph/9906268] [INSPIRE].
[33] S. Bejar, F. Dilme, J. Guasch and J. Sol`a, Higgs boson flavor changing neutral decays into
bottom quarks in supersymmetry, JHEP 08 (2004) 018 [hep-ph/0402188] [INSPIRE].
– 52 –
JHEP05(2012)015
[20] LHC/LC Study Group collaboration, G. Weiglein et al., Physics interplay of the LHC and
the ILC, Phys. Rept. 426 (2006) 47 [hep-ph/0410364] [INSPIRE].
[34] A.M. Curiel, M.J. Herrero and D. Temes, Flavor changing neutral Higgs boson decays from
squark Gluino loops, Phys. Rev. D 67 (2003) 075008 [hep-ph/0210335] [INSPIRE].
[35] A. Curiel, M. Herrero, W. Hollik, F. Merz and S. Pe˜
naranda, SUSY electroweak one loop
contributions to flavor changing Higgs-Boson decays, Phys. Rev. D 69 (2004) 075009
[hep-ph/0312135] [INSPIRE].
[36] T. Hahn, W. Hollik, J. Illana and S. Pe˜
naranda, Interplay between H → b¯
s and b → sγ in the
MSSM with Non-Minimal Flavor Violation, hep-ph/0512315 [INSPIRE].
[38] S. Dittmaier, G. Hiller, T. Plehn and M. Spannowsky, Charged-Higgs collider signals with or
without flavor, Phys. Rev. D 77 (2008) 115001 [arXiv:0708.0940] [INSPIRE].
[39] J. Cao et al., SUSY-induced FCNC top-quark processes at the Large Hadron Collider,
Phys. Rev. D 75 (2007) 075021 [hep-ph/0702264] [INSPIRE].
[40] G. Degrassi, P. Gambino and P. Slavich, SusyBSG: a Fortran code for BR(B → Xs γ) in the
MSSM with Minimal Flavor Violation, Comput. Phys. Commun. 179 (2008) 759
[arXiv:0712.3265] [INSPIRE].
[41] G. Degrassi, P. Gambino and P. Slavich, QCD corrections to radiative B decays in the MSSM
with Minimal Flavor Violation, Phys. Lett. B 635 (2006) 335 [hep-ph/0601135] [INSPIRE].
[42] G. Isidori and P. Paradisi, Hints of large tan β in flavour physics,
Phys. Lett. B 639 (2006) 499 [hep-ph/0605012] [INSPIRE].
[43] G. Isidori, F. Mescia, P. Paradisi and D. Temes, Flavour physics at large tan β with a
Bino-like LSP, Phys. Rev. D 75 (2007) 115019 [hep-ph/0703035] [INSPIRE].
[44] S. Baek, T. Goto, Y. Okada and K.-I. Okumura, Muon anomalous magnetic moment, lepton
flavor violation and flavor changing neutral current processes in SUSY GUT with
right-handed neutrino, Phys. Rev. D 64 (2001) 095001 [hep-ph/0104146] [INSPIRE].
[45] J. Foster, K.-I. Okumura and L. Roszkowski, Probing the flavor structure of supersymmetry
breaking with rare B-processes: a beyond leading order analysis, JHEP 08 (2005) 094
[hep-ph/0506146] [INSPIRE].
[46] F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, A complete analysis of FCNC and
CP constraints in general SUSY extensions of the Standard Model,
Nucl. Phys. B 477 (1996) 321 [hep-ph/9604387] [INSPIRE].
¯d mixing and the Bd → J/ψKs asymmetry in general SUSY models,
[47] D. Becirevic et al., Bd -B
Nucl. Phys. B 634 (2002) 105 [hep-ph/0112303] [INSPIRE].
[48] L.J. Hall, V.A. Kostelecky and S. Raby, New flavor violations in supergravity models,
Nucl. Phys. B 267 (1986) 415 [INSPIRE].
[49] J. Gunion and H.E. Haber, Higgs bosons in supersymmetric models. 1,
Nucl. Phys. B 272 (1986) 1 [Erratum ibid. B 402 (1993) 567] [INSPIRE].
[50] J. K¨
ublbeck, M. B¨
ohm and A. Denner, FeynArts: computer algebraic generation of Feynman
graphs and amplitudes, Comput. Phys. Commun. 60 (1990) 165 [INSPIRE].
[51] T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3,
Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
– 53 –
JHEP05(2012)015
[37] J. Cao, G. Eilam, K.-I. Hikasa and J.M. Yang, Experimental constraints on stop-scharm
flavor mixing and implications in top-quark FCNC processes, Phys. Rev. D 74 (2006) 031701
[hep-ph/0604163] [INSPIRE].
[52] The FeynArts website, http://www.feynarts.de/.
[53] T. Hahn and M. P´erez-Victoria, Automatized one loop calculations in four-dimensions and
D-dimensions, Comput. Phys. Commun. 118 (1999) 153 [hep-ph/9807565] [INSPIRE].
[54] T. Hahn and C. Schappacher, The implementation of the minimal supersymmetric Standard
Model in FeynArts and FormCalc, Comput. Phys. Commun. 143 (2002) 54
[hep-ph/0105349] [INSPIRE].
¯ → Xs γ, Nucl. Phys. B 611 (2001) 338
[55] P. Gambino and M. Misiak, Quark mass effects in B
[hep-ph/0104034] [INSPIRE].
[57] W. Porod and F. Staub, SPheno 3.1: extensions including flavour, CP-phases and models
beyond the MSSM, arXiv:1104.1573 [INSPIRE].
[58] B. Allanach et al., The Snowmass points and slopes: benchmarks for SUSY searches,
Eur. Phys. J. C 25 (2002) 113 [hep-ph/0202233] [INSPIRE].
[59] O. Buchmueller et al., Supersymmetry and Dark Matter in light of LHC 2010 and Xenon100
data, Eur. Phys. J. C 71 (2011) 1722 [arXiv:1106.2529] [INSPIRE].
[60] S. Bertolini, F. Borzumati, A. Masiero and G. Ridolfi, Effects of supergravity induced
electroweak breaking on rare B decays and mixings, Nucl. Phys. B 353 (1991) 591 [INSPIRE].
[61] P.L. Cho, M. Misiak and D. Wyler, KL → π 0 e+ e− and B → Xs ℓ+ ℓ− decay in the MSSM,
Phys. Rev. D 54 (1996) 3329 [hep-ph/9601360] [INSPIRE].
[62] G. Degrassi, P. Gambino and G. Giudice, B → Xs γ in supersymmetry: large contributions
beyond the leading order, JHEP 12 (2000) 009 [hep-ph/0009337] [INSPIRE].
¯ → Xs+d γ CP asymmetry as a probe for new
[63] T. Hurth, E. Lunghi and W. Porod, Untagged B
physics, Nucl. Phys. B 704 (2005) 56 [hep-ph/0312260] [INSPIRE].
[64] M. Ciuchini, G. Degrassi, P. Gambino and G. Giudice, Next-to-leading QCD corrections to
B → Xs γ: Standard Model and two Higgs doublet model, Nucl. Phys. B 527 (1998) 21
[hep-ph/9710335] [INSPIRE].
[65] L.J. Hall, R. Rattazzi and U. Sarid, The top quark mass in supersymmetric SO(10)
unification, Phys. Rev. D 50 (1994) 7048 [hep-ph/9306309] [INSPIRE].
[66] M.S. Carena, D. Garcia, U. Nierste and C.E. Wagner, b → sγ and supersymmetry with large
tan β, Phys. Lett. B 499 (2001) 141 [hep-ph/0010003] [INSPIRE].
[67] M.S. Carena, D. Garcia, U. Nierste and C.E. Wagner, Effective Lagrangian for the t¯bH +
interaction in the MSSM and charged Higgs phenomenology, Nucl. Phys. B 577 (2000) 88
[hep-ph/9912516] [INSPIRE].
[68] G. Isidori and A. Retico, Scalar flavor changing neutral currents in the large tan β limit,
JHEP 11 (2001) 001 [hep-ph/0110121] [INSPIRE].
[69] A.J. Buras, P.H. Chankowski, J. Rosiek and L. Slawianowska, ∆Md,s , B 0 d, s → µ+ µ− and
B → Xs γ in supersymmetry at large tan β, Nucl. Phys. B 659 (2003) 3 [hep-ph/0210145]
[INSPIRE].
[70] G. Isidori and A. Retico, Bs,d → ℓ+ ℓ− and KL → ℓ+ ℓ− in SUSY models with nonminimal
sources of flavor mixing, JHEP 09 (2002) 063 [hep-ph/0208159] [INSPIRE].
– 54 –
JHEP05(2012)015
[56] S. AbdusSalam et al., Benchmark models, planes, lines and points for future SUSY searches
at the LHC, Eur. Phys. J. C 71 (2011) 1835 [arXiv:1109.3859] [INSPIRE].
[71] A. Crivellin and U. Nierste, Chirally enhanced corrections to FCNC processes in the generic
MSSM, Phys. Rev. D 81 (2010) 095007 [arXiv:0908.4404] [INSPIRE].
[72] A. Crivellin and U. Nierste, Supersymmetric renormalisation of the CKM matrix and new
constraints on the squark mass matrices, Phys. Rev. D 79 (2009) 035018 [arXiv:0810.1613]
[INSPIRE].
[73] K.-I. Okumura and L. Roszkowski, Weakened Constraints from b → sγ on supersymmetry
flavor mixing due to next-to-leading-order corrections, Phys. Rev. Lett. 92 (2004) 161801
[hep-ph/0208101] [INSPIRE].
[75] J. Foster, K.-I. Okumura and L. Roszkowski, New Higgs effects in B-physics in
supersymmetry with general flavour mixing, Phys. Lett. B 609 (2005) 102 [hep-ph/0410323]
[INSPIRE].
[76] Heavy Flavor Averaging Group collaboration, D. Asner et al., Averages of b-hadron,
c-hadron and τ -lepton properties, arXiv:1010.1589 [INSPIRE].
¯ → Xs γ: current status, Acta Phys. Polon. B 40 (2009) 2987
[77] M. Misiak, B
[arXiv:0911.1651] [INSPIRE].
0
[78] P.H. Chankowski and L. Slawianowska, Bd,s
→ µ− µ+ decay in the MSSM,
Phys. Rev. D 63 (2001) 054012 [hep-ph/0008046] [INSPIRE].
[79] C. Bobeth, T. Ewerth, F. Kr¨
uger and J. Urban, Enhancement of
¯d → µ+ µ−) )/B(B
¯s → µ+ µ−) ) in the MSSM with Minimal Flavor Violation and large
B(B
tan β, Phys. Rev. D 66 (2002) 074021 [hep-ph/0204225] [INSPIRE].
[80] K. Babu and C.F. Kolda, Higgs mediated B 0 → µ+ µ− in minimal supersymmetry,
Phys. Rev. Lett. 84 (2000) 228 [hep-ph/9909476] [INSPIRE].
[81] C. Bobeth, T. Ewerth, F. Kr¨
uger and J. Urban, Analysis of neutral Higgs boson
¯ → Kℓ+ ℓ− , Phys. Rev. D 64 (2001) 074014
contributions to the decays Bs → ℓ+ ℓ− and B
[hep-ph/0104284] [INSPIRE].
[82] CMS and LHCb collaborations, Search for the rare decay Bs0 → µ+ µ− at the LHC with the
CMS and LHCb experiments. Combination of LHC results of the search for Bs → µ+ µ−
decays, CMS-PAS-BPH-11-019, CERN, Geneva Switzerland (2011) [INSPIRE].
[83] A.J. Buras, Flavour theory: 2009, PoS(EPS-HEP 2009)024 [arXiv:0910.1032] [INSPIRE].
[84] D. Becirevic, V. Gim´enez, G. Martinelli, M. Papinutto and J. Reyes, B parameters of the
complete set of matrix elements of ∆B = 2 operators from the lattice, JHEP 04 (2002) 025
[hep-lat/0110091] [INSPIRE].
[85] D. Becirevic, V. Gim´enez, G. Martinelli, M. Papinutto and J. Reyes, Combined relativistic
and static analysis for all ∆B = 2 operators, Nucl. Phys. Proc. Suppl. 106 (2002) 385
[hep-lat/0110117] [INSPIRE].
[86] A.J. Buras, M. Jamin and P.H. Weisz, Leading and next-to-leading QCD corrections to ǫ
¯ 0 mixing in the presence of a heavy top quark,
parameter and B 0 -B
Nucl. Phys. B 347 (1990) 491 [INSPIRE].
[87] E. Golowich, J. Hewett, S. Pakvasa, A.A. Petrov and G.K. Yeghiyan, Relating Bs mixing and
Bs → µ+ µ− with new physics, Phys. Rev. D 83 (2011) 114017 [arXiv:1102.0009] [INSPIRE].
– 55 –
JHEP05(2012)015
[74] K.-I. Okumura and L. Roszkowski, Large beyond leading order effects in b → sγ in
supersymmetry with general flavor mixing, JHEP 10 (2003) 024 [hep-ph/0308102] [INSPIRE].
[88] E. Lunghi, Flavor physics in the LHC era: the role of the lattice, talk given at Lattice 2011,
http://tsailab.chem.pacific.edu/lat11/plenary/lunghi/lunghi-lattice2011.pdf, July 11–16
2011.
[89] J. Gunion, H. Haber, G. Kane and S. Dawson, The Higgs hunter’s guide, Addison-Wesley,
Boston U.S.A. (1990).
[90] A. Dabelstein, Fermionic decays of neutral MSSM Higgs bosons at the one loop level,
Nucl. Phys. B 456 (1995) 25 [hep-ph/9503443] [INSPIRE].
[91] A. Dabelstein, The one loop renormalization of the MSSM Higgs sector and its application to
the neutral scalar Higgs masses, Z. Phys. C 67 (1995) 495 [hep-ph/9409375] [INSPIRE].
– 56 –
JHEP05(2012)015
[92] A. Denner, Techniques for calculation of electroweak radiative corrections at the one loop
level and results for W physics at LEP-200, Fortsch. Phys. 41 (1993) 307
[arXiv:0709.1075] [INSPIRE].

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